away, around 1–10 pc from the MBH. These putative IMBHs are supposedly
harboured in a handful of compact gaseous clouds, whose measured velocity
dispersion is so high to suggest the presence in their centres of point-like objects with
masses in the range $10^4–10^5 M_\odot$ (Oka et al. 2017; Takekawa et al. 2019, 2020).
However, depending on their orbital properties, a population of IMBHs lurking at the
Galactic Centre would affect significantly the motion of S-stars (Deme et al. 2020b)
and the structure of the nuclear star cluster (Mastrobuono-Battisti et al. 2014).

The detection of such heavy IMRs by LISA would have huge implications for our
understandings of IMBH formation and evolution.

### 3.2.3.3 Light IMRs in stellar clusters and dwarf nuclei
We can also imagine the formation of light IMRs, where the IMBH is the more massive partner in a merger with a stellar mass BH. At first glance, one might expect these to be simply scaled down versions of the EMRI problem with a central MBH. However, the challenge of understanding stellar dynamics with an IMBH is that we cannot assume that the IMBH is fixed at the centre of the system, as we do with MBH in galactic nuclei. The wandering of the IMBH makes it demanding, to say the least, to attempt an analytical study, and hence we have to resort to numerical simulations to get an idea of what could be the IMRI event rate and the characteristic properties. Globular clusters and dwarf nuclei may also present quite different dynamical scenarios (higher escape velocities due to dark matter but lower stellar densities in dwarf galaxies, for example).

The first dynamical simulation addressing the evolution of a globular cluster
harbouring an IMBH which successfully led to the formation of an IMRI was done
by Konstantinidis et al. (2013). In this work they employed a direct-summation
*N*-body code with relativistic corrections as presented by and a live treatment of the
relativistic recoil. They find that IMBHs with masses 500–1000 $M_\odot$ merge with
stellar-mass BHs and escape the host globular cluster due to the low escape velocity
of the system. The IMBH is in a binary in almost all cases. The companion is a
stellar-mass BH of mass ~20–26 $M_\odot$, and semi-major axis of about 5–7 AU. Later,
Leigh et al. (2014) found similar results for this mass range. In their simulations, the
heaviest stellar-mass BH forms a tight binary with the IMBH in the system. The work
of Haster et al. (2016a) is basically a reproduction of Konstantinidis et al. (2013), but
using a different numerical scheme. This is interesting because it validates of the
findings of Konstantinidis et al. (2013). Later, MacLeod et al. (2016b) explore lighter
mass ranges for the IMBH, with masses at most of 150 $M_\odot$. They confirm that the
IMBH has a bound companion most of the time, with the probability distribution
function for the semi-major axis maximised at 2 AU.

Recently, Pestoni et al. (2021) performed a series of Fokker-Planck simulations to
explore the occurrence of light IMRs around IMBHs of $10^5 M_\odot$ residing at the
centre of massive star-forming clumps in high-redshift galaxies, finding event rates of
$10^{-8}–10^{-7}$ year$^{-1}$, depending on the assumptions for the initial inner density profile.

The IMRI rate from globular clusters and detectable by LISA depends intrinsically
on a number of unknown quantities, namely the fraction of clusters capable of
nursing the IMBH seed and growth (Portegies Zwart and McMillan 2002; Giersz
et al. 2015), the amount of stellar-mass BHs and other compact objects lurking in the IMBH closest vicinity (MacLeod et al. 2016b; Arca Sedda et al. 2019a), the number of times the same IMBH can pair in an IMRI (MacLeod et al. 2016b), and the probability that upon merger an IMBH is ejected from the parent cluster due to anisotropic GW emission (Holley-Bockelmann et al. 2008; Fragione et al. 2018; Arca Sedda et al. 2021a). Depending on all these quantities, LISA might be able to detect 0.01–60 IMRIs from globular clusters per year out to redshift $z \sim 2$ (Arca Sedda et al. 2020a, 2021a). The number of light IMRIs from massive star-forming clumps detectable by LISA falls in the same ballpark: by integrating their computed IMRI event rate over $z = 1–3$, when clumpy galaxies are more numerous, Pestoni et al. (2021) computed that LISA should be able to detect $\sim 2$ IMRIs per year, conservatively assuming that one star-forming clump per clumpy galaxy hosts a central IMBH. IMRIs with IMBHs in the mass range between $10^2 M_{\odot}$ and a few $10^3 M_{\odot}$ might be detected with LISA and provide advanced warning to ground-based detectors with a precision up to a second (Amaro-Seoane 2018a).

### 3.2.3.4 The contribution of LISA to the physics of heavy- and light-IMRI formation in gas-poor environments
In the processes we have just described with regard to the formation and evolution of heavy- and light IMRIs in gas-poor scenarios, our astrophysical theory so far only provides loose guidance about how the physics of the assembly process connects to the characteristics of this population. Current models suggest, though, that understanding heavy IMRIs will be an important discriminator between hierarchical formation models.

IMRIs provoke more open questions than EMRIs, as the theoretical frameworks required to address IMRI formation are even more complex, due to the mobility of an IMBH, as compared to an MBH. Thus, many approximations used in relaxation theory to predict the gas-poor dynamics of EMRI formation cannot be applied to an otherwise similar IMBH system. In addition, there are several plausible channels of IMRI formation that involve theoretical frameworks that have far larger uncertainties than relaxation theory.

As for galactic nuclei assembly, several uncertainties might affect the formation of such heavy IMRIs, namely the number of clusters capable of reaching the galactic centre, the probability of IMBH formation, the relation between IMBH formation and the host cluster properties. Theoretical models suggest that star cluster infall and dispersal could bring 1-50 IMBHs in the MBH vicinity (Portegies Zwart et al. 2006; Mastrobuono-Battisti et al. 2014; Arca-Sedda and Gualandris 2018; Arca-Sedda and Capuzzo-Dolcetta 2019; Leveque et al. 2022; Fragione 2022). If these models prove right, the delivery of IMBHs can give rise to up to 0.001–0.03 Gpc$^{-3}$ year$^{-1}$ (Arca-Sedda and Capuzzo-Dolcetta 2019; Fragione 2022), corresponding roughly to 1 event per year within redshift 0.5-3 (Fragione 2022). Interestingly, the possible ejection of the IMRI product from the parent nucleus owing to GW radiation could account for up to $10^5$ Gpc$^{-3}$ MBH wandering outside their host galaxies at redshift < 1 (Arca-Sedda and Capuzzo-Dolcetta 2019).

Finally, less consideration has been given to possible light IMRI rates in dwarf nuclei, although the presence of at least some IMBH in dwarf galaxies is more secure
than in globular clusters. This is an important open question, especially since IMRI
detection in dwarf nuclei could enable us to probe IMBH in *quiescent* dwarf galaxies.
This in turn leads to a better understanding of the formation of seed MBHs. The
detection of even a single light IMRI with LISA would incredibly improve our
knowledge of how and where IMBHs form and grow.

3.2.4 Formation of EMRIs and IMRIs in gas-rich galactic nuclei: AGN discs

**3.2.4.1 Gas-rich dynamics: active galactic nuclei** As discussed above, many galactic nuclei harbor a dense nuclear cluster of stellar-origin objects surrounding an MBH. Above we considered the state of the NSC as a result of stellar evolution, dynamical friction, secular evolution and minor mergers (Morris 1993; Antonini 2014; Generozov et al. 2018). Further complicating our picture, however, are the existence of AGN, which occur when low angular momentum gas forms a disc that accretes onto the MBH. As a result, a fraction of the nuclear cluster will end up embedded in the AGN disc via coincident orbits or capture (Syer et al. 1991; Artymowicz et al. 1993). While one might anticipate that AGN would be a subdominant mechanism for producing EMRIs or IMRIs, given that AGN represent only a fraction of all galactic nuclei (or, more likely, a relatively brief episode or series of episodes in the life of any given galactic nucleus), the presence of gas qualitatively changes the dynamics in the NSC.

Prograde orbiters embedded in a disc are expected to have their eccentricities rapidly damped (Ward 1988; Tanaka and Ward 2004; Cresswell et al. 2007; Bitsch and Kley 2010), although there is a strong dependence on the details and resolution of gas-flow on horse-shoe orbits (Bitsch and Kley 2010). For plausible AGN disc densities, gas dynamical cooling dominates over spherical component dynamical heating, so prograde orbiters should experience very rapid (<0.1 Myr) eccentricity damping (McKernan et al. 2012; Kennedy et al. 2016; MacLeod and Lin 2020).

The majority of NSC objects begin as inclined orbiters not coincident with the disc. Stars experience geometric drag and BHs experience dynamical drag as they pass through the disc, causing a significant portion to be captured within a plausible range of AGN disc lifetimes (0.1–100 Myr). Stellar and BH orbiters not coincident with the disc experience geometric and dynamical drag forces, damping first the orbital eccentricity, followed by orbital inclination (Bitsch and Kley 2011; Just et al. 2012; Kennedy et al. 2016; Panamarev et al. 2018; MacLeod and Lin 2020; Fabj et al. 2020). Fabj et al. (2020) find O(10%) of prograde NSC BH are captured within a Sirkó and Goodman (2003) type disc, for typical disc lifetimes, ignoring accretion. A much smaller fraction are captured by lower-density type Thompson et al. (2005) AGN discs. Stellar objects are primarily captured by the disc at small radii, losing around an order of magnitude in semi-major axis, whereas BHs are captured across the full range of disc radii but rapidly get delivered to the innermost disc (Fabj et al. 2020). The possibility of the accumulation of BHs at small radii across disc models has significant implications for the LIGO–Virgo merger detection rate (Fabj et al. 2020), but also for the possible EMRI/IMRI production rate.

The accumulation of prograde BH from the NSC in the inner disc leads to high
interaction cross sections at low relative velocity. All embedded stellar-origin objects
on prograde orbits should undergo mass-dependent Type I migration due to torques
from gas at Lindblad resonances and co-rotating gas (Tanaka et al. 2002), enhancing
pile-up of BHs in inner AGN discs, leading to a high merger rate (McKernan et al.
2012, 2014; Bartos et al. 2017; Stone et al. 2017b).

**3.2.4.2 Heavy IMRs in AGN** The rate of change of surface density in AGN disc models implies that we should expect the occurrence of locations in the discs where the outward and inward migration torques cancel (Bellovary et al. 2016). At such so-called migration traps, the local merger rate is significantly enhanced and IMBHs with masses ~$10^3 M_\odot$ can quickly (1 Myr) be produced (Secunda et al. 2019, 2020; Yang et al. 2019; McKernan et al. 2020b). The IMBH-formation merger GW190521 detected by LIGO-Virgo (Abbott et al. 2020d) consisted of two BHs in the upper mass gap with mis-aligned spins, suggestive of a merger in a dynamically rich, deep gravitational potential well. If we assume this merger happened at a migration trap then the rate of such mergers inferred by LIGO-Virgo is ~$0.7 \text{ Gpc}^{-3} \text{ year}^{-1}$ The LIGO Scientific Collaboration et al. (2020). From McKernan et al. (2020a), assuming $O(15)$ mergers at migration traps over a Myr disc lifetime, and an AGN fraction of $O(1\%)$ of galactic nuclei (quasars and the brightest Seyfert nuclei), we find ~$1 \text{ Gpc}^{-3} \text{ year}^{-1}$ mergers at migration traps, consistent with the observed rate of GW190521-like events. All of these effects taken together enhance stellar BH merger rates and encourage the formation of IMBH in AGN discs. IMBH formed in this manner automatically create an IMBH-MBH binary, which will typically decay due to GW emission on timescales of a few hundred Myr (Bellovary et al. 2016).

**3.2.4.3 Light IMRs in AGN** A large IMBH sitting in an AGN disc migration trap is an excellent site for the creation of light IMRs—less massive BH will be delivered to the IMBH from more distant regions of the disc via disc migration torques, with an approximate merger rate of $O(1) \text{ Gpc}^{-3} \text{ year}^{-1}$ (McKernan et al. 2020a), assuming Sirko and Goodman (2003) type AGN disc models. LISA can detect IMBHs of several hundred $M_\odot$ out to ~10 Gpc. So for migration trap mergers with an IMBH of few hundred $M_\odot$, this suggests LISA can detect $O(10^3) \text{ year}^{-1}$ light IMRI mergers from AGN discs. If migration traps are less common in AGN discs, the maximum IMBH masses from bulk disc mergers are much smaller ~$10^2 M_\odot$—and migration torques could drive objects rapidly onto the MBH, typically creating EMRIs with small eccentricities and inclination angles.<fnref n="14" />

**3.2.4.4 EMRs in AGN** One likely exception to the small (*e*, *i*) expectation due to AGN gas damping comes from retrograde orbiters. Approximately half of the initial (NSC) population that is geometrically coincident with the disc, should lie on retrograde orbits (Ivanov et al. 2015). Migration torques on retrograde orbiters in AGN discs are a small fraction of that on prograde orbiters (McKernan et al. 2014). However, retrograde orbiters experience eccentricity pumping at apocenter which

<footnote n="14">¹⁴ In providing fuel to the MBH, this process may contribute to explaining the $M_{bh}-\sigma$ relation (Miralda-Escudé and Kollmeier 2005).</footnote>
rapidly drives them to very high eccentricities (Dunhill et al. 2013; Teyssandier and Ogilvie 2016, 2019; MacLeod and Lin 2020) and increases the decay rate of the semi-major axis (Secunda et al. 2021). As a result we expect a significant population of retrograde orbiters in the innermost AGN disc. This population could yield a very high EMRI rate. The eccentricities of EMRs from this population depend on the masses of the retrograde orbiters, since the GW circularization rate increases with mass (Peters 1964a) and the eccentricity driving of the gas decreases with mass (Secunda et al. 2021). For example, a $10 M_{\odot}$ retrograde orbiter will likely have an eccentricity over 0.9 when it inspirals, whereas a $50 M_{\odot}$ orbiter will circularize before inspiraling, making it much easier for LISA to detect over many cycles. However, the higher rate of eccentricity driving of lower mass retrograde orbiters also implies that their EMRI rates will be higher. Therefore, unlike prograde orbiters whose orbits tend to be circularized by the gas disc, retrograde orbiters could commonly produce highly eccentric EMRs, though this effect is strongest at high (AGN disc) gas density and for low mass BH.

There may also be observable EM signals associated with smaller BHs in AGN discs. Stellar-mass BH binaries (BH+BHs) can merge at high rates in AGN discs (McKernan et al. 2012, 2014; Bartos et al. 2017). Under these circumstances, since the BH+BH is surrounded by gas, there will always be an EM counterpart. Indeed, a candidate EM counterpart to GW190521 has recently been suggested in an AGN (Graham et al. 2020). Several key questions underpin the search for EM counterparts to BH+BH mergers in AGN discs: Is the EM counterpart detectable through a potentially large optical depth? Is the emission completely outshined by the AGN emission, and on what timescale? Does the radiation from the BH reduce the EM emission?

At merger, a remnant stellar-mass BH formed from the BH+BHs recoils with a kick velocity $v_k$ depending on the mass asymmetry and spin orientations of the progenitors. In an AGN disc, gas at distance $R_{\text{bound}} < GM_{\text{BH+BH}}/v_k^2$ is bound to the merged BH+BH and attempts to follow the kicked merger product. In doing so, it collides with surrounding disc gas and a shock luminosity emerges on a time-scale of ~20 days, with a bound gas energy of about $10^{45}$ erg (depending on the BH+BH and gas properties). After the kicked stellar-mass BH has shed the bound gas, the passage of the stellar-mass BH through the gas in the accretion disc produces a shocked Bondi drag tail (e.g. Ostriker 1999). This tail both decelerates the stellar-mass BH and accretes onto it, generating a potentially high luminosity. In order for enough radiation to escape to make for a bright flare against the AGN, a jet or collimated outflow is required. Such jets may also produce detectable X-ray or gamma-ray signatures.

Alternatively, stellar-mass BHs in AGN discs may undergo substantial accretion even prior to merger (Yang et al. 2020b), which can lead to e.g., longer term X-ray emission. This scenario largely depends on the poorly understood accretion efficiency and gap opening by stellar-mass BHs in AGN discs. We recommend that future simulations of hyper-Eddington accretion establish whether there is an upper limit to accretion which can choke off jet formation and launching. This will help establish luminosity upper limits on any flares that originate from kicked stellar-
mass BH mergers in AGN discs and can guide searches for EM counterparts from AGN discs. We also recommend simulations of lightcurves from false-positive flaring events such as SNe and TDEs breaking out from within the AGN disc, or lightcurves of micro-lensing events.

### 3.2.4.5 In situ formation of stars in AGN discs: a special population of EMRs
AGN discs are known to be prone to gravitational (Toomre) instability in the outer regions, and expected to form stars vigorously (Shlosman and Begelman 1989; Goodman 2003; Levin 2007; Nayakshin et al. 2007). This expectation is supported by observations of nearby stellar discs in the nucleus of the MW (Levin and Beloborodov 2003) and M31 (Bender et al. 2005) which, due to their large masses and orbital configurations, can be interpreted as remnants of a prior accretion episode from a gaseous disc (Levin 2007). Numerous observational studies also suggest a broader connection between AGN activity and nuclear starbursts (e.g., Davies et al. 2007; Wild et al. 2010; Ishibashi and Fabian 2016), although whether this connection is causal remains uncertain, given that AGN feeding occurs on scales difficult to resolve (≤ parsecs) and is often obscured (Alexander and Hickox 2012). Theoretically, the stars formed in the outskirts of AGN discs are expected to be unusually massive because the disc material is much hotter and denser than star-forming regions in the galactic ISM. Once formed, a population of stars embedded in a gas disc will undergo a stellar evolution that is notably altered by their environment (Cantiello et al. 2021). Throughout their lifetime stars can grow by accretion, becoming even more massive (Goodman and Tan 2004; Davies and Lin 2020). In addition to experiencing drag and dynamical friction, they will excite perturbations in the disc that will exert torques on their orbit, typically causing inward migration (as discussed above). Furthermore, we expect an increased number of binaries within the stellar population in the AGN disc (e.g., Alexander et al. 2008), which can become harder due to disc-satellite interactions (e.g., Baruteau et al. 2011; Arca Sedda 2020a). All this makes fertile ground for forming BH remnants in the disc, which can subsequently encounter each other in migration traps (described above) or migrate to the inner regions of the disc where their evolution becomes GW dominated. Initial estimates show that this process may be an efficient method for feeding MBHs at early times in a way that is not Eddington-limited (Dittmann and Miller 2020).

Future numerical studies can help us narrow down the vast parameter space of accretion disc structures as a function of relevant parameters such as MBH mass, accretion rate, gas supply or redshift. As we improve our understanding of AGN discs, more investigations are needed to fully understand how embedded stars and BHs evolve over a range of system parameters and disc properties. Including more detailed physics (such as disc instabilities and stochastic torques, radiation transport or feedback from accretion, to name a few) may change the evolutionary outcome of these sources, the predicted rates and characteristic properties, as well as the precise waveform signatures and whether or not they are distinguishable amongst formation channels. At the same time, gas-embedded EMRs/IMRs also present a powerful opportunity to probe AGN properties in regions that are historically electromagnetically unresolvable, either with deviations in the GW waveforms that correlate with
properties of the gas (see Sect. 3.4) or with populations statistics, if multiple events are detected. This is in addition to the possibility for multimessenger astrophysics with associated EM counterparts (e.g., variability in emission, see Sect. 3.3).

As a distinguishing characteristic between various formation channels, here we quote approximate expectations of eccentricity and inclination at late stages of the inspiral, which we define as approaching the central MBH ISCO. Gas-driven, prograde EMRs are expected to have low $e$ ($e \lesssim 0.01$) given that circularization by gas and GWs is efficient, but there remains a possibility of disc-driven eccentricity pumping for relatively massive secondaries or IMRs (e.g., D'Angelo et al. 2006). These sources are also likely fully embedded in the disc with low inclination—hence if the orientation of the disc aligns with the spin of the central MBH (which may be true to varying degrees, see Volonteri et al. 2013), we expect the spins of the binary components to be closely aligned. Gas driven, *retro*-grade EMRs, on the other hand, may reach very high eccentricities even at the ISCO ($e \lesssim 0.9$, Secunda et al. 2021). They should also retain a low inclination, although the spin alignment of the BHs will depend on the accretion history of the embedded BH which remains to be investigated across the full range of parameter space. These estimates will be further constrained with future work that includes more realistic disc modelling and treatment of gas dynamics and accretion onto embedded BHs.

### 3.2.4.6 The contribution of LISA to the physics of formation of EMRs and IMRs in gas-rich galactic nuclei
From what we have presented, we can derive that depending on the (highly uncertain) duty cycle of AGN, an IMBH-MBH binary could correspond to a heavy IMRI in nearly every galactic nucleus. However, the absence of any such detection with LISA would seriously constrain the existence of a migration trap in a generic AGN disc. The absence of a migration trap implies the absence of strong changes in the disc surface density gradient and thus tightly constrains the transition between the radiation pressure and gas pressure dominated regions of AGN discs.

As we have mentioned, there is a clear correlation between the dynamical parameters of EMRs formed in AGN discs and their detection. Thus the rate of EMRs detected by LISA can put strong constraints on the populations and dynamics we expect to live in innermost AGN discs, a system lurking in a region inaccessible to spatially resolved EM observations.

Prediction of rates and precise characteristics of disc-embedded EMRs/IMRs is a multi-faceted problem that relies on details of gravitational instability, stellar evolution, nonlinear gas dynamics, and accretion physics. Given that MBHs spend 1–10% of their evolution in an AGN phase (Shankar et al. 2013; Pardo et al. 2016) we expect at minimum the same fraction of EMRs to occur in gas-rich environments. In-situ star formation likely leads to a population of compact remnants in addition to those that are captured from the nucleus. Thus we expect that dense accretion discs in near-Eddington AGN may not only boost EMRI rates, but also produce a population that is uniquely characteristic: with low eccentricity and (some degree of) spin alignment with the central MBH. These ($e, i$) expectations are strongest for EMRs from objects formed in-situ. A single EMRI with low
eccentricity will indicate a current (or recent) interaction with gas, showing that the host galaxy harbors an active (or recently active) MBH. The precise parameters (e.g. eccentricity, secondary BH mass) are intimately connected to prior evolution of the accretion disk, and thus these measurements will give constraints on the efficiency of gas-driven circularization and accretion disk structure. If a larger population of such EMRIs is detected, the distribution of orbital characteristics will tell us about the diversity of AGN disks. Rates and orbital characteristics will constrain various aspects of accretion onto MBHs—for example, the number of detected events will inform how many nearby AGN host disk-embedded BHs. If the number is high, it will challenge accretion models. Just as the migration rates of stars and compact remnants depend sensitively on characteristics of the disk (Baruteau and Masset 2013; Duffell 2015), the secondary masses will be a consequence of BH accretion or hierarchical mergers, all of which are tied to disk models. Overall, measurements that shed light on accretion disk structure and prevalence will improve our constraints on AGN duty cycles and MBH growth.

### 3.2.5 Alternative formation scenarios

#### 3.2.5.1 XMRIs
The possibility of observing an EMRI at our own galactic centre when LISA flies is basically zero, since the rates for an EMRI formed via relaxation in a MW-like galaxy are at most about $10^{-6}$ year$^{-1}$ (e.g., Amaro-Seoane 2018b, 2021). This means that about once every million years a stellar-mass BH plunges through the event horizon of our central MBH. Since the lifetime of LISA will be only a few years, the probability of detecting one at our galactic centre is negligible.

We can however find another class of EMRIs at our galactic centre. It has been recently put forward (Amaro-Seoane 2019) that substellar objects, in particular brown dwarfs, stand very high chances of being in band of the detector when it is launched. The reason for this is very simple: these substellar objects have mass ratios of about $q \sim 10^{-8}$ as compared to the central MBH, Sgr A*. Such XMRIs (extremely large mass ratio inspirals) can therefore cover up to $\sim 10^8$ cycles before crossing the event horizon, since the number of cycles is roughly inversely proportional to the mass ratio. This means that they stay in band for millions of years. About $2 \times 10^6$ year before merger they have an SNR at the galactic centre of 10. Later, $\sim 10^4$ year before merger, the SNR reaches several thousands, i.e. they are at the level of the loudest MBH mergers. At the last stages of their evolution, some $\sim 10^3$ year before the merger, they can reach SNR as high as a few $10^4$ (Amaro-Seoane 2019; Gourgoulhon et al. 2019; Barack and Cutler 2004b).

The work of Amaro-Seoane (2019) predicts that at any given moment there should be of the order of $\ge 5$ XMRI that are highly eccentric and are located at higher frequencies, and about $\ge 15$ are circular and are at lower frequencies. The mass ratio for an XMRI is about three orders of magnitude smaller than that of stellar-mass BH EMRIs. Since backreaction depends on $q$, the orbit closely follows a standard geodesic, which means that many approximations work better in the calculation of
the orbit. XMRIs can be sufficiently loud so as to track the systematic growth of their SNR, which can be high enough to bury that of MBH binaries.

In addition, there are also plunge events during the formation of inspiralling sources. The GWs from low mass objects (brown dwarfs, primordial BHs, etc.) plunging into the central MBH are burst signals. For LISA, the SNRs of these bursts are quite high if they happen in our Galaxy. However, the event rates are estimated as ~0.01 year⁻¹ for the Galaxy. If we are lucky, this kind of very extreme mass-ratio burst will offer a unique chance to reveal the nearest MBH and nucleus dynamics. The event rate could be as large as 4–8 year⁻¹ within 10 Mpc, and because the signal is strong enough for observations by space-borne detectors, there is a good chance of being able to use these events to probe the nature of neighbouring BHs (Berry and Gair 2013b). This kind of burst sources are called XMRBs (extreme mass ratio bursts) (Han et al. 2020).

### 3.2.5.2 Binary and multiple EMRIs
Recent theoretical studies pointed out the existence of a new type of EMRI in which the small body is a stellar-mass BH+BH. Such a triple system could form either due to tidal capture of a BH+BH by a MBH (Addison et al. 2019; Chen and Han 2018) or the formation and migration of a BH+ BH in the accretion disc of an AGN (Chen et al. 2019a). While the latter channel is considered to be more effective than the former one, both channels could deliver BH+BHs to a distance as small as tens of gravitational radii of the central MBH. As a result, the binary, as a single entity, spirals into the MBH due to GW radiation. For this reason, the source is referred to as a binary-EMRI, or b-EMRI.

The uniqueness of the b-EMRI lies in the fact that it simultaneously emits two kinds of GWs. One in the LISA band, due to the orbital motion of the BH+BH around the MBH, and the other in the ground-based detector band, when the BH+ BH coalesces due to the tidal perturbation by the MBH. A coordinated observation by LISA and ground-based detectors would allow us to identify such interesting sources and, more importantly, constrain several aspects of fundamental physics to a precision more than one order of magnitude better than the current limit, including the loss of rest mass due to GW radiation, the recoil velocity of the merging BH+ BH, and the dispersion of GWs of difference frequencies (Han and Chen 2019). Due to the merger of the BH+BH, the remnant will obtain a recoil velocity which may be up to a few thousand km s⁻¹. This sudden kick will induce a glitch on the waveform of a b-EMRI (see Fig. 33).

### 3.2.5.3 Supernova-driven EMRIs
In addition to the aforementioned mechanisms, an EMRI can also be generated via the supernova explosion that accompanies the formation of a CO. When this happens, the velocity of the compact object gets almost instantaneously significantly perturbed, so that the compact object settles on a brand new trajectory: the timescale for the CO to coalesce with the MBH via GWs on this new orbit may be shorter than the timescale for two-body relaxation to perturb it, so that the CO is bound to evolve into an EMRI. Focusing on the Galactic Centre environment, Bortolas and Mapelli (2019) showed that one supernova out of 10⁴–10⁵ occurring within the star forming structures present about the MW MBH will give
<figure><img src="image_35.png" /><figcaption>**Fig. 33** Comparing the waveforms of the EMRIs with (red, solid) and without (blue, dashed) a glitch. The MBH has a mass of $M = 10^6 M_{\odot}$ and a spin parameter of 0.9. The total mass of the BH+BH is $m = 20 M_{\odot}$, and *D* refers to the luminosity distance. In this example, the centre of mass of the stellar-mass BH+BH initially is moving inside the equatorial plane of the MBH with an orbital eccentricity of *e* = 0.7 and a semilatus rectum of *p* = *R*(1 − *e*²) = 17*r*<sub>g</sub>. At the time *t* = 0 a kick to the centre-of-mass velocity of the binary happens, in the polar direction and with a magnitude of 1500 km s⁻¹. As a result, the orbital parameters change to *p* = 16.9990*r*<sub>g</sub> and *e* = 0.7019, and the orbital plane of the EMRI becomes inclined by *t* = 0.5233° relative to the equatorial plane of the MBH</figcaption></figure>

rise to a supernova-driven EMRI. This result, coupled with the expected frequency of core-collapse supernovae explosions occurring in the Galactic Centre, implies a frequency of supernova-driven EMRIs up to 10⁻⁸ year⁻¹, i.e., an EMRI rate that is comparable or only mildly lower than the one associated to the standard two-body relaxation process (Bortolas and Mapelli 2019).

### 3.2.5.4 The contribution of LISA to relativistic stellar dynamics and supernovae rates
What we have described about XMRIs allows us to understand that these might be envisaged as a double-edged sword. From the one side we have a promising and strong source of GWs from an extreme-mass ratio which is easy to model. On the other hand they might pose a problem because their SNRs (as high as 10⁴ for 1-year observation if one resides in the Galactic Center) are such that can bury binaries of MBHs. Also, if they are present in most nuclei harbouring MBHs in the range of LISA, they might interact with EMRIs or even scatter them off from their inspiraling orbit. The detection of XMRIs will allow us to infer information on relativistic astrodynamics impossible to obtain otherwise.

LISA can distinguish b-EMRIs from normal EMRIs by detecting the GWs from the small binary black hole months to years prior to its coalescence around the SMBH. If the frequency of the GWs from the small binary matches a fundamental frequency of the SMBH, the SMBH could be resonantly excited and the EMRI waveform could contain an enhanced quasi-normal mode (Cardoso et al. 2021). Moreover, the binarity of the small body also induces an additional phase shift to the EMRI waveform, which can be used to identify b-EMRIs as well (Chen and Zhang 2022).

Regarding the detection of EMRIs formed via supernova, this result calls for a more extended analysis, in order to investigate this process in a wider range of galaxy environments and star-formation rates, and exploring in more detail the waveform signatures associated to this kind of EMRI compared to other EMRI formation
mechanisms. In this framework, LISA will thus help us shed light on the rates of supernovae near MBHs via the detection of EMRIs.

## 3.3 Multimessenger prospects

**Coordinators:** Giuseppe Lodato and Martina Toscani

**Contributors:** Pau Amaro Seoane, Jillian Bellovary, Stefano Bianchi, Saavik Ford, Barry McKernan, Giuseppe Lodato, Tom Kimpson, Scott Noble, Martina Toscani, Kinwah Wu, Ziri Younsi and Silvia Zane

We outline a variety of proposed EM counterparts of EMRIs and IMRIs, including TDE (in multiple configurations), AGN-related signatures, and pulsar EMRIs.

### 3.3.1 Tidal disruption events

TDEs (Carter and Luminet 1983; Rees 1988; Phinney 1989; Rossi et al. 2021, for a recent review) can be considered a particular type of EMRI, where the star is disrupted by the MBH tides during the first passage at the pericenter. For this to happen, the pericenter radius should be smaller (cf. Ryu et al. 2020) than the *tidal radius*

$$r_t \approx R_* \left( \frac{M_{\bullet}}{M_*} \right)^{1/3}, \qquad (18)$$

where $R_*$ and $M_*$ are the stellar radius and stellar mass respectively, while $M_{\bullet}$ is the BH mass; $r_t$ is usually a factor of 10–20 times the MBH Schwarzshild radius. For MBH masses larger than $\sim 10^8 M_{\odot}$ (10$^9 M_{\odot}$ for rapidly spinning BHs), the tidal radius is within the event horizon and no TDEs can happen.

TDEs are very luminous events over a broad range of EM bands. The first EM observations occurred in 1990s thanks to the ROSAT survey (Bade et al. 1996; Komossa and Greiner 1999; Grupe et al. 1999; Greiner et al. 2000), which detected some bright flares from the cores of non-AGN galaxies. Since then, the number of X-ray detections has increased. These observations seem to be in agreement with the theoretical expectation of X-ray emission from an accretion disc (e.g., Ulmer 1999; Auchettl et al. 2017; Lodato and Rossi 2011). Initially these flares are powered by a near-Eddington accretion, then the luminosity decreases over a period from months to years (Saxton et al. 2020, and references therein). Over the last decade, also a growing number of optical TDEs has been detected. It remains unclear what processes are at the origin of this optical emission. Some hypotheses concern the shocks from self-crossing debris (Piran et al. 2015; Shiokawa et al. 2015) or reprocessing in an outflow (e.g., Strubbe and Quataert 2009; Lodato and Rossi 2011; Metzger and Stone 2016). Detailed reviews of optical TDEs are found in Wevers et al. (2019) and van Velzen et al. (2020). A small fraction of these jetted events has also shown significant radio emission (Alexander et al. 2020, and references therein).
Furthermore, TDEs emit GWs. Their GW emission is produced by three different varying quadrupoles: (a) the star-BH quadrupole (Kobayashi et al. 2004; Toscani et al. 2022; b) the stellar internal quadrupole (Guillochon et al. 2009; Stone et al. 2013) and (c) the quadrupole of the compact torus formed after disruption (e.g., van Putten 2001, 2002; Kiuchi et al. 2011; Toscani et al. 2019; van Putten et al. 2019). The dominant contribution is the first term, that could be well described as a GW burst with strain (Kobayashi et al. 2004; Toscani et al. 2022)

$$h \approx \beta \times \frac{r_s r_s^*}{r_t d} \approx \beta \times 2 \times 10^{-22} \left(\frac{M_*}{M_{\odot}}\right)^{4/3} \left(\frac{M_{\bullet}}{10^6 M_{\odot}}\right)^{2/3} \left(\frac{R_*}{R_{\odot}}\right)^{-1} \left(\frac{d}{16 \, \text{Mpc}}\right)^{-1}, \quad (19)$$

and an associated Keplerian frequency of

$$f \approx \frac{\beta^{3/2}}{2\pi} \left( \frac{GM_{\bullet}}{r_t^3} \right)^{1/2} \approx \beta^{3/2} \times 10^{-4} \, \text{Hz} \times \left( \frac{M_*}{M_{\odot}} \right)^{1/2} \left( \frac{R_*}{R_{\odot}} \right)^{-3/2}. \quad (20)$$

In the above formulas we have introduced the Schwarzschild radius of the BH, $r_s = 2r_g$, the Schwarzschild radius of the star, $r_s^*$, and the penetration factor

$$\beta = \frac{r_t}{r_p}, \quad (21)$$

where $r_p$ is the pericentre distance. A library of gravitational waveforms from TDEs has been presented in Toscani et al. (2022), generated using a general relativistic smoothed particle hydrodynamic code, PHANTOM (Liptai and Price 2019). To date, this numerical study models the star as a polytropic sphere with index $\gamma = 5/3$, but we expect the strain to have a dependence on the internal structure of the star. This dependence needs to be further investigated.

The expected strain for a Sun-like star being disrupted by a $10^6 M_{\odot}$ MBH at 15 Mpc distance (assuming $\beta = 1$) is $h \sim 10^{-22}$, with a frequency $f \sim 10^{-4}$ Hz. The two other contributions are expected to have similar frequency but are scaled down by some (two-five) orders of magnitude.

Pfister et al. (2022) estimate the rate of TDEs which could be observed with different instruments. They find for LISA it is unlikely to detect GWs from TDEs, unless BHs are surrounded by particularly massive stars. The next generation of detectors beyond LISA should however be able to detect GW from TDEs up to cosmological redshifts $z \ge 1$.

An interesting signal to study is the GW background from the entire cosmic population of TDEs. Details on this signal and its derivation may be found in Sect. 3.5.1.

### 3.3.1.1 TDEs outside galactic nuclei
In the majority of TDE studies the MBH that disrupts a star is implicitly taken to be the nuclear BH of the galaxy. The Swift transient source AT2018cow has many characteristics resembling a TDE (Kuin et al. 2019), but peculiarly the source is not located at the nuclear region of its suspected
host galaxy Z 137-068. This leads to consideration of whether it is a TDE, with an MBH disrupting a WD, i.e. WD-TDE (Han and Fan 2018), or alternatively a violent stellar explosion. The question is now: Can TDE involving a wandering<fnref n="15" /> MBH occur? This question would be answered if there are mechanisms to populate a galaxy with massive BHs of non-stellar nature. Stellar systems are not stationary structures. A stellar cluster can dissolve on timescales of 10–100 Myr (e.g., Gieles and Bastian 2008). Globular clusters can survive longer but they can also be disrupted (e.g., Belokurov et al. 2006; Wan et al. 2020) or dissolved (see Baumgardt 2009). Similarly, dwarf galaxies can be disrupted and dissolved (e.g., Li et al. 2018; Sanders et al. 2018) when they encounter and are accreted by a larger galaxy. The nuclear BHs, if present in these stellar systems, would be dispersed into the interstellar space of the cannibal galaxy. Stars are also carried along into the interstellar space by these BHs, and some of them will eventually spiral into their carrier BH and become a TDE.

### 3.3.2 Electromagnetic counterparts of light IMRs in AGN discs

Most BHs that merge in AGN discs (including IMBH-BH mergers) are expected to experience a GW recoil kick at the moment of merger with speeds $v_k$ of up to a few hundred km s⁻¹. Such merger kicks would happen for any comparable system, with or without gas; however, the consequences of such kicks for a gas-embedded merger may include a detectable EM counterpart. In an AGN disc, gas within

$$R_{\text{bound}} < \frac{GM_{\text{BH+BH}}}{v_k^2} \qquad (22)$$

is bound to the merged BH+BH and attempts to follow the kicked merger product. In doing so, it collides with surrounding disc gas and, as long as the disc is geometrically thin or optically thin, a shock luminosity can emerge on a timescale $t_{\text{bound}} = R_{\text{bound}}/v_k = GM_{\text{BH+BH}}/v_k^3$ (McKernan et al. 2019) or

$$t_{\text{bound}} \sim 20 \left( \frac{M_{\text{BH+BH}}}{100 M_{\odot}} \right) \left( \frac{v_k}{200 \text{ km s}^{-1}} \right)^{-3} \text{day.} \qquad (23)$$

The total energy delivered to the bound gas is $E_{\text{bound}} = (1/2)M_{\text{bound}}v_k^2 = (3/2)Nk_B T_{\text{bound}}$ where $M_{\text{bound}} = Nm_H$ is the mass of the bound gas expressed as N atoms of Hydrogen (mass $m_H$), $k_B$ is the Boltzmann constant, and $T_{\text{bound}}$ is the average temperature of the post-shock gas. This energy is

$$E_{\text{bound}} = 3 \times 10^{45} \left( \frac{\rho}{10^{-10} \text{ g cm}^{-3}} \right) \left( \frac{M_{\text{BH+BH}}}{100 M_{\odot}} \right)^3 \left( \frac{v_k}{200 \text{ km s}^{-1}} \right)^{-4} \text{erg}, \qquad (24)$$

and the resulting average hot spot temperature is

<footnote n="15">¹⁵ See Sects. 2.2.1.1, 2.2.2.4 and 2.2.3.</footnote>
$$T_{\text{bound}} \sim 1.8 \times 10^6 \left( \frac{v_k}{200 \text{ km s}^{-1}} \right)^2 \text{K.} \qquad (25)$$

The resulting UV/optical flare occurs between $t = [0, t_{\text{ram}}]$, has an average (low) luminosity $E_{\text{bound}}/t_{\text{ram}}$ and a shape given by $\sin^2(\pi t/2t_{\text{ram}})$.

Once the kicked BH leaves behind originally bound gas, the disc gas it passes through is accelerated around the BH, producing an asymmetric low angular momentum Bondi tail inside the stagnation point (e.g., Ostriker 1999; Antoni et al. 2019). This tail both acts as a drag on the BH and accretes onto it. The Bondi–Hoyle–Lyttleton luminosity is $L_{\text{BHL}} = \eta \dot{M}_{\text{BHL}} c^2$ where $\eta$ is the radiative efficiency and

$$\dot{M}_{\text{BHL}} = \frac{4\pi G^2 M_{\text{BH}}^2 + B H \rho}{v_{\text{rel}}^3}, \qquad (26)$$

with $v_{\text{rel}} = v_k + c_s$ and $c_s$ is the gas sound speed. In principle, hyper-Eddington accretion is allowed by this process. This should cause trapping of emergent radiation, unless a collimated outflow allows radiation to escape. However, if the kicked merger product travels out of the dense disc midplane into a more tenuous disc atmosphere, such signatures may be bright enough to be detected even against bright AGN hosts (Graham et al. 2020).

### 3.3.3 FeKα lines (or other EM signatures) as probes of small separation MBH–IMBH binaries

The relativistically broadened component of the fluorescent FeKα line (centered around 6.4–7 keV source-frame) is believed to be a probe of material in the innermost accretion disc (Nandra et al. 1997; Fabian et al. 2000; Reynolds and Nowak 2003). EMRIs and heavy IMRIs pre-merger will disrupt the flow of gas in the innermost disc, yielding flicker in the innermost disc (EMRIs) or carving gaps or a central cavity (MBH–MBH, MBH–IMBH binaries) with minidiscs. The resulting re-arrangement of gas yields signatures prior to GW merger events which may be detectable in the broad FeKα with high-throughput X-ray telescopes like Athena. A gap-opening secondary IMBH close to the primary MBH will leave an imprint in the broad component of the FeKα emission line, which varies in a unique and predictable manner (McKernan et al. 2013).

### 3.3.4 EMRIs containing a pulsar

Pulsars are spinning NSs, mainly identified by their radio observations. To date there are about 2800 known pulsars in the Galaxy (Cameron et al. 2020), of which about 160 are found to be associated with globular clusters (see the ATNF pulsar catalogue, Manchester et al. 2005). Among these radio pulsars, more than 170 are MSPs (e.g., Levin et al. 2013), and the majority of them actually reside in globular clusters. Although the number of known MSPs is growing, most MSPs are yet to be discovered. It has been suggested that the total population number of MSPs in the
MW could be about 100,000 or even more (Levin et al. 2013). Radio pulsar timing is a relatively mature technique in astronomy, as researchers have accumulated experience in pulsar research over decades. The current developments in instrumentation and search techniques will enable us to detect radio pulsars outside the MW (Keane et al. 2015), and the detection range will be further extended by the time LISA is operating.

EMRs containing a radio pulsar are a special class of GW sources with a guaranteed EM counterpart, if they are close enough. The presence of a MSP provides researchers with several advantages to study these EMRI systems and their associated physics. NSs have a small mass range centred around $1.4 M_{\odot}$. Knowing the mass and the spin of one component in the EMRI system reduces the parameter space, hence easing the computational demands in the template matching and searching for establishing their GW properties, whereas other EMRI systems would require the determination of the system parameters simultaneously, relying solely on the GW signals. The availability of the EM signals with measurements at high precision will give another advantage. Both pulsar timing observations and GW experiments can obtain measurements to high precision; the accuracy and precision of pulsar timing is among the highest achievable in astrophysical time-domain analysis (Hartnett and Luiten 2011). Radio pulsar timing and GW experiments employ different analysis techniques. As such, the orbital and spin dynamics, as well as the system parameters which they determine, will be independent, thereby giving us a means to understand certain systematic properties in the statistical and data analyses.

### 3.3.4.1 The contribution of LISA to multimessenger science
The event rate of the potential emission of GWs by extended stars approaching a MBH will provide us with additional information about tidal disruption events. This combined with EM detections will deliver much more precise catalogues of disruptions and, within some limits, information about the star and MBH which is inaccessible via traditional telescopes.

Regarding TDEs outside galactic nuclei, a WD-TDE system would emit GWs (Han and Fan 2018) as well as bursts of EM radiation (Kuin et al. 2019). With the additional constraints provided by the GW observations, it would easily resolve the dispute about certain candidate TDE sources, such as AT 2018cow (Kuin et al. 2019; Perley et al. 2019).

From what we have explained about EM counterparts of light IMRs, we can conclude that a population of stellar-origin BH+BH in the LISA band that harden into the ground-based GW detector band in AGN discs can yield potentially detectable optical/UV counterparts. IMBH-BH and IMBH-IMBH binary mergers in AGN discs are likely to occur at migration traps in the inner disc (Bellovary et al. 2016), so kicked merger products remain bound to the MBH. The kicked BH must splash back down into the AGN disc possibly yielding a repeat flare on half the orbital timescale. An off-center luminous flare should be detectable as an asymmetry in broad optical lines as the broad line region responds to non-central illumination (McKernan et al. 2019).
Double relativistic FeKα lines may be detectable from binary mini-disc emission, allowing us to localize LISA sources well before merger (Sesana et al. 2012). The barycenter of a MBH binary will lie outside the event horizon of the primary BH for modest values of mass ratio and binary separation. Analogous to the radial velocity method of planet detection, whereby the wobble of a star indicates the presence of a nearby Jupiter-sized planet, the radial velocity of the primary BH around the binary barycenter can leave a tell-tale oscillation in the broad component of FeKα emission (McKernan and Ford 2015). Such oscillations are detectable by Athena for binaries with mass ratios $q \ge 0.01$, at binary separations of up to $O(10^2 rg)$. Both the general-relativistic and Lense-Thirring precession of the periapse of the secondary orbit imprint a detectable modulation on these oscillations (McKernan and Ford 2015). Athena is likely to detect $O(30)$ FeKα broad lines at sufficient statistical significance in local AGN to carry out tests for ripples and oscillations of such binaries. $O(100)$ AGN may have broad FeKα components that will allow us to search for double components (McGee et al. 2020). Hence, the input from LISA and Athena can be compounded to extract information about the separation of heavy IMRs.

LISA detection of EMRIs with a MSP will provide opportunities to investigate a variety of fundamental issues in gravitational physics. This is rooted in the extreme mass ratio between the MSP and the BH (the MSP being a test mass) and the ultra-fast rotation of the MSP (the MSP being an extreme gyro and a stable time-keeper). More specifically, the spin-spin, spin-orbit, and the spin-curvature interactions (Chicone et al. 2005; Iorio 2012; Remmen and Wu 2013; Singh et al. 2014) between the MSP and the BH will manifest in the spin and orbital dynamics of the MSP (Li et al. 2019; Kimpson et al. 2020b), which will in turn modify the pulsar timing signals via modification of the pulse period and the pulse arrival time (Kimpson et al. 2019a, 2020a). Together with the information extracted from the GWs generated by the EMRI, researchers will be able to investigate how EM waves propagate in a non-vacuum space time (Kimpson et al. 2019b) or in a slightly perturbed space time, as well as having the opportunity to gain some understanding of certain fundamental issues, such as the gravitational self-force (e.g., Barack and Pound 2019) in GW sources.

## 3.4 Environmental effects on waveforms

**Coordinators:** Alvin Chua, Alejandro Torres-Orjuela and Lorenz Zwick

**Contributors:** Pau Amaro-Seoane, Manuel Arca Sedda, Emanuele Berti, Xian Chen, Alvin Chua, Andrea Derdzinski, Kyriakos Destounis, Wen-Biao Han, Kostas Kokkotas, Cole Miller, Scott Noble, Arthur Suvorov, Alejandro Torres-Orjuela and Lorenz Zwick

We know that EMRI/IMRI events can form in a variety of interesting astrophysical environments. Some of these environments may leave detectable imprints on the waveforms measured by LISA (though detecting the imprints may be challenging).
We outline a wide variety of environmental effects, including gas-driven effects and many-body effects. Some effects may be degenerate with one another or with deviations from general relativity, even when an effect is detectable. Fortunately, complex dynamics sometimes lend themselves to breaking degeneracies through, e. g., Doppler effects. We also use this section to address the possibility of detecting (and extracting astrophysical information from) the EMRI background, and possibly detecting the signatures of chaotic systems. Finally, we specifically consider the degeneracies between environmental effects and PN/self-force effects.

### 3.4.1 Gas torques

For gas-embedded EMRs/IMRs, gas torques can speed up or slow down an inspiral while it is in the LISA band. The magnitude of the torques will scale with the disc density, and the precise value and direction of the torque will depend on the mass of the inspiralling CO and disc properties. The effect is several orders of magnitude weaker than GWs in this regime, but even a small dephasing over several thousand cycles may accumulate to a detectable phase shift (up to a few radians), depending on the density of the environment, as shown in analytic work (Yunes et al. 2011a; Kocsis et al. 2011; Barausse et al. 2014) as well as more recently in 2D hydrodynamical simulations (Derdzinski et al. 2019, 2021). Accretion discs in bright AGN are expected to be thin and dense, but their inner regions are hot and radiation pressure-dominated. Analytical estimates of densities in the inner regions of such discs from simple models predict surface densities varying from ~$10–10^7$ g cm$^{-2}$ (Shakura and Sunyaev 1973; Frank et al. 2002). The wide range arises from our uncertainty on how viscosity scales with the (gas or total) pressure. State-of-the art 3D global magnetohydrodynamical disc simulations (Jiang et al. 2016, 2019; Jiang and Blaes 2020) suggest that densities are between these values, somewhat closer to the lower end, which could make gas dephasing too small to detect, but this may change as we continue to explore the parameter space of MBH masses.

Whether or not this effect is *detectable* will depend on the density of the environment as well as the SNR of the source. An EMRI embedded in a Shakura–Sunyaev alpha-disc with $\Sigma \sim 10^2$ g cm$^{-2}$ can accumulate a phase shift up to $\lesssim 10^{-2}$ radians within 4 years, whereas if embedded in a beta-disc would dephase over $10^1–10^2$ radians over 4 years (Derdzinski et al. 2021). IMRs, due to their higher mass, accumulate higher SNR and also feel stronger torques (since they scale with secondary mass), making them ideal events for producing detectable gas signatures.

Whether or not this effect is *distinguishable* from other waveform deviations will depend on how well one can measure the phase shift and how this accumulates as the frequency evolves. Simulations suggest that the torque can be approximated by simple analytical formulae—torques are within an order of magnitude of the Type I torque derived by Tanaka et al. (2002), although variability in the torque can arise for sufficiently massive secondaries ($q \gtrsim 10^{-3}$). This means that one can estimate how the deviation accumulates with frequency, assuming we know the disc density profile, and if $\ddot{f}$ can be measured from the GW data, degeneracies between the GW waveform distortions due to disc torques versus parameter variations or other effects
can be disentangled in principle. In practice this may prove difficult given that the small mass ratio of these sources (and expected low eccentricity) means they will chirp slowly, and will more likely appear as near-continuous wave sources within a few year observation.

### 3.4.2 Many-body interactions

#### 3.4.2.1 XMRIs and EMRIs
As explained above, we expect a handful of XMRIs to be present in our own Galactic Centre (Amaro-Seoane 2019). Since these systems are so loud, reaching SNRs of up to ~20,000, in principle we should be able to detect them in nearby galaxies harbouring MBHs in the mass range $10^5-10^7 M_\odot$ (i.e. nuclei for which the relaxation time is below a Hubble time Amaro-Seoane 2021, 2018b; Preto 2010). Farther away XMRI systems with much lower SNR will not be detected. However, XMRIs pose a problem for normal EMRI systems: Since XMRIs live in band for millions of years, and the estimated event rate for an EMRI is between $10^{-5}-10^{-6} \text{ year}^{-1}$, the possibility that an EMRI encounters an XMRI on its way to the MBH is non-negligible.

#### 3.4.2.2 EMRIs interacting with a perturbing star
Although unlikely, it is not ruled out that a star can be located close to an EMRI inspiraling towards the central MBH. In the work of Amaro-Seoane et al. (2012) the authors derive the shortest radius from the MBH within which one might expect to have at least one star. Then they run direct-summation N-body simulations with relativistic corrections following the first implementation as presented in Kupi et al. (2006) and find that periapsis shift along with gravitational-radiation effects induce non-determinism in the evolution of the EMRI. This means that for two identical dynamical setups of an EMRI system with a perturbing star located at a distance of about ~5$a_{\text{EMRI}}$, with $a_{\text{EMRI}}$ the semi-major axis of the EMRI, small changes of any dynamical parameter induces a different evolution of the EMRI. The presence of a perturbing star, therefore, can be misinterpreted as a deviation of general relativity, and this should be taken into account in the development of data analysis algorithms.

#### 3.4.2.3 Binary-EMRIs
The formation rate of b-EMRIs for BH+BHs tidally captured by MBHs is equivalent to $(10^{-5}-10^{-4}) \text{ Gpc}^{-3} \text{ year}^{-1}$ in the pessimistic case and 0.1 $\text{ Gpc}^{-3} \text{ year}^{-1}$ in the most optimistic one. However, due to the non-negligible lifetime of b-EMRIs, within a spherical volume of $1 \text{ Gpc}^3$ (corresponding to a radial distance of about 600 Mpc), there are, on average, about 0.02–20 b-EMRIs expected during LISA’s mission duration.

The most exciting property of b-EMRIs is that they are multi-band GW sources, i.e., radiate low and high frequency GWs synchronously (Addison et al. 2019). Though the time scale of BH+BH merger is much shorter than the inspiral of binary into the MBH, LISA and the ground-based detectors (LIGO etc.) may observe the b-EMRI at the same time. The high-frequency GWs could be redshifted because they are generated close to a MBH (Chen et al. 2019a), providing an opportunity of studying the propagation of GWs in the regime of strong gravity.
### 3.4.3 Moving sources

Almost all astrophysical objects are moving relative to us and GWs sources are no exception. The motion of the centre of mass of a source is often related to the properties of its environment, e.g., the orbital motion induced by the interaction with other bodies (Wen 2003; McKernan et al. 2012; Antonini and Perets 2012; Naoz 2016; Arca Sedda 2020a; Stone et al. 2017b; Bartos et al. 2017; Tagawa et al. 2020a) and the motion of its host system like the peculiar velocity of galaxies (Zinn and West 1984; Bahcall 1988; Carlberg et al. 1996; Springel et al. 2001; Scrimgeour et al. 2016; Colin et al. 2017; Girardi et al. 1996; Ruel et al. 2014). Therefore, the detection of velocity can provide valuable and versatile information about the sources environment and its host system.

In the case of a constant velocity the Doppler effect changes the observed GW frequency $f_{\text{obs}}$ by a factor (Chen et al. 2019a)

$$f_{\text{obs}} = f(1+z)^{-1}, \qquad (27)$$

and its derivative, $\dot{f}_{\text{obs}}$, by the same factor squared

$$\dot{f}_{\text{obs}} = \dot{f}(1+z)^{-2}. \qquad (28)$$

Here, $f$ and $\dot{f}$ being, respectively, the GW frequency and its derivative in the source's rest frame. When only considering the dominant mode of GWs, these shifts lead to a wrong estimation for the sources actual chirp mass $M$,

$$M_{\text{obs}} = M(1+z), \qquad (29)$$

and actual luminosity distance, $d_{\text{L}}$,

$$d_{\text{obs}} = d_{\text{L}}(1+z), \qquad (30)$$

thus fundamentally affecting our interpretation of the source. An analogous effect appears for the cosmological redshift of GWs. Although the latter one is considered in current GW models and detections, the same is in general not true for the effect of velocity (Abbott et al. 2019).

If the velocity of the source varies in time, the previous picture changes significantly. The Doppler effect induces a time-dependent phase shift, proportional to the velocity of the source along the line of sight, which can be detected when having accurate models of the evolution of the phase of the source and the velocity profile (Inayoshi et al. 2017b; Meiron et al. 2017; Wong et al. 2019a). For a LISA mission of 4 years probably no accelerated sources could be detected. However, the number of detections could be increased to up to 3 when conducting joint measurements with a ground-based detector. For a LISA mission of 10 years up to 40 accelerated sources could be detected by LISA alone and up to 103 when conducting joint measurements with an earth based detector (Tamanini et al. 2020). Moreover, the aberration of the GWs rays affects the line of sight thus inducing an additional phase shift (Torres-Orjuela et al. 2020). The magnitude of the aberrational phase shift
is of the same order as the Doppler phase shift but proportional to the components of
the velocity perpendicular to the line-of-sight. Therefore, considering the total phase
shift, i.e. Doppler effect plus aberrational phase shift, the SNR required for the
detection of the acceleration could be reduced by a factor of up to 1.8, thus allowing
the detection of up to 5.8 times more sources (Torres-Orjuela et al. 2020).

3.4.4 Dark matter as an environmental effect

The density profile of dark matter halos has a cusp at the centre of galaxies because of the large potential well there (Kuhlen et al. 2012). If a MBH resides at the centre of the galaxy, the strong gravity could lead to a significant increase of density in the central region and create a spike, which enhances the dark matter annihilation rate (Gondolo and Silk 1999; Sadeghian et al. 2013). Similarly, IMBHs may have a smaller dark matter spike (Zhao and Silk 2005; Bertone et al. 2005). The gravitational potential of the dark matter could impact the evolution of an EMRI/IMRI, particularly where enhancement of the density occurs due to spikes or a superradiant instability, leading to a detectable signature (Eda et al. 2013; Yue and Han 2018; Hannuksela et al. 2020). However, dynamical events such as mergers of host galaxies can weaken the dark matter cusp (Ullio et al. 2001; Merritt et al. 2002; Merritt 2004; Bertone and Merritt 2005), which makes its presence harder to detect.

3.4.5 Astrophysical chaos

Owing to the huge mass disparity for the objects involved in an EMRI, the dynamics
of the companion body can be modelled as a point particle traversing the
gravitational field of the (super-massive) primary to high accuracy. Characteristics of
GWs emitted during the inspiral are therefore dominated by the particulars of the
metric geometry of the primary. Fundamental symmetries, or the absence thereof,
associated with the spacetime geometry can therefore be probed by LISA. For a
primary which is both stationary and axisymmetric—properties expected of
astrophysically stable BHs—the energy and one component of the angular
momentum of a companion are both constants of motion with respect to the orbital
dynamics. The companion’s Hamiltonian, $H \sim g_{\mu\nu}p^\mu p^\nu$ for momentum $p$, provides a
third constant of motion. In *four* spacetime dimensions, however, having only *three*
conserved quantities implies that the equations of motion are not Liouville integrable
(Contopoulos 2002). In this context, a non-integrable system exhibits chaotic orbital
phenomena, as is familiar from the three-body problem in (post-)Newtonian gravity
(Huang and Wu 2014). The Kerr spacetime, which uniquely represents stable BHs in
general relativity, also however admits a rank-two Killing tensor, which provides a
fourth constant of motion in the form of the Carter constant (Carter 1968). This
implies the absence of astrophysical chaos in EMRIs within general relativity, at least
for companions which are not themselves spinning rapidly (Kiuchi and Maeda 2004;
Lukes-Gerakopoulos et al. 2014; Piovano et al. 2020; Zelenka et al. 2020).

If, however, high-energy corrections to the field equations present themselves in
nature, the particulars of gravitational collapse (e.g., Cembranos et al. 2012) and
accretion (e.g., Harko et al. 2010) may be such that a non-Kerr object resides within galactic centres or elsewhere. There are many ways in which a hypothetical departure from a Kerr description may manifest within the spacetime metric, such as those described in Johannsen (2013). One interesting possibility is that the Carter symmetry is broken, thereby giving rise to a non-Kerr object, as opposed to a deformed-Kerr body which still forbids chaotic phenomena even if the spacetime is not exactly Kerr (Papadopoulos and Kokkotas 2018; Destounis et al. 2020).

In general, sections of the inspiral that behave as bound orbits can be characterised by both radial ($\omega_r$) and angular ($\omega_\theta$) libration frequencies, which describe the rate of transition from the periastron to the apastron of the orbit and longitudinal oscillations about the equatorial plane, respectively (Contopoulos 2002). On the other hand, classical results from dynamical systems theory infer that small islands of stability (Birkhoff islands) form around periodic orbits in the phase space of non-integrable dynamical systems (Arnold 1978). When an inspiralling orbit crosses an island, the ratio $\omega_r/\omega_\theta$, which defines what is called the *rotation curve*, remains constant, while otherwise it behaves monotonically as a function of radius. Absence of islands therefore implies an everywhere monotonic rotation curve, while the dynamics display transient plateau features for non-Kerr spacetimes when orbits intersect with an island (Apostolatos et al. 2009).

Several studies have shown that these transient plateaus also introduce features into the gravitational waveforms which are, in principle, discernible from deformed-Kerr features (Apostolatos et al. 2009; Lukes-Gerakopoulos et al. 2010; Contopoulos et al. 2011; Cárdenas-Avendaño et al. 2018; Destounis et al. 2020; Lukes-Gerakopoulos and Witzany 2021). Figure 34 shows the fundamental frequency evolution of a gravitational waveform, associated with a particular non-Kerr spacetime (see Destounis et al. 2020, 2021; Destounis and Kokkotas 2021 for details). However, non-integrable perturbations in the Arnold (1978) sense of the particle Hamiltonian may also arise due to environmental effects within general relativity (Cardoso et al. 2022). For instance, $N > 2$-body interactions (as in the Newtonian case; see also Sect. 3.4.2) (Barausse et al. 2007; Amaro-Seoane et al. 2012) or significant internal spins in the companion (Kiuchi and Maeda 2004; Lukes-Gerakopoulos et al. 2014) can induce chaos.

### 3.4.6 Environment versus PN/self-force degeneracies

The secular evolution of EMRI orbital elements is intimately connected with the phase and the shape of the GWs that will be measured by LISA. In vacuum this evolution is fully described by general relativity, and it can in principle be computed to arbitrary precision with approximation schemes such as perturbation theory (an expansion in the small mass ratio $q$ of the binary) or the PN expansion (an expansion in the small parameter $v/c$, where $v$ is the orbital velocity and $c$ is the speed of light). At leading order in perturbation theory, the system can be described as a point-like particle moving in a geodesic orbit around the large BH. The energy flux can be computed either numerically or analytically to varying degrees of accuracy. For example, Munna (2020) computed the energy radiated from eccentric orbits around nonrotating BHs up to 19PN order. Fujita (2015) computed the energy flux from a
<figure><img src="image_36.png" /><figcaption>**Fig. 34** Periodogram of the fundamental frequency of a gravitational waveform associated with an inspiral on a particular non-Kerr spacetime. A sudden jump in the frequency evolution appears when the inspiralling object crosses a Birkhoff island, which is associated with a valley in the amplitude of the signal's frequencies (white line)</figcaption></figure>

particle in *circular* orbit around a rotating BH up to 11PN order. Sago and Fujita (2015) computed the expansion for *eccentric* orbits around rotating BHs up to 4PN, and up to order $e^6$ in a small-eccentricity expansion. The PN expansion is an asymptotic series, and it is known to converge quite slowly for EMRIs (Yunes and Berti 2008; Zhang et al. 2011). At higher orders in $q$, interaction of the particle with its own gravitational perturbation gives rise to gravitational self-force, which drives the radiative evolution of the orbit, and whose effects can be accounted for by order in $q$ (Barack and Pound 2019).

EMRIs are however by necessity embedded in astrophysical environments, and as such it is likely that their secular evolution will differ from the pure vacuum case. Of all the environmental factors, gravitational torques from accretion flows are likely to be most the significant (Cardoso and Maselli 2020). Obtaining realistic estimates for the influence of accreting gas on the orbital evolution and phase of the binary is difficult because accretion dynamics is a largely unexplored 3D magnetohydrodynamics problem over a large dynamic range. The best theoretical prediction for the impact of gas dynamics on IMRI/EMRI phase errors are from 2D viscous hydrodynamics (Derdzinski et al. 2019, 2021). Phase errors grow with the surface density of the accretion disc. The sign in the tidal torque, i.e. whether the separation increases or decreases, depends on the mass ratio, the strength of the viscosity (the $\alpha$ parameter), and the rate of inspiral (Derdzinski et al. 2021). Some parameters even experience stochastic variability in the sign of the tidal torque for EMRIs; such
variability would be in stark contrast with secular phase errors coming from truncating the PN expansion. Phase errors due to non-stochastic effects, however, will likely be comparable to PN errors for some disc configurations and orbital separations (Barausse et al. 2014; Cardoso and Maselli 2020; Annulli et al. 2020). Understanding whether these situations are likely requires a combination of results from 2D viscous simulations (cf. Derdzinski et al. 2019, 2021) with EM surveys of AGN discs, population synthesis modelling of AGN/binary discs (e.g., Krolik et al. 2019), and estimates of the distribution of the observed binary parameters for LISA. Since all EMRI/IMRI simulations have used 2D Newtonian viscous hydrodynamics, it will be interesting to see how these results change when using more realistic 3D general-relativistic magnetohydrodynamical simulations. Unfortunately, performing a series of simulations with O(2000) orbits, which seems to be required to reach a steady-state in 2D simulations, is computationally prohibitive at present.

Another obvious effect that can spoil the vacuum evolution of an EMRI is the influence of a third gravitational body. In the case of a hierarchical triple, two effects can take place. First, the influence of the perturber can produce a shift in the binding energy of the inner binary. Will (2014) showed that this shift is constant even if the inner binary undergoes perihelion advance. Second, if the perturber is sufficiently inclined it can induce von Zeipel–Kozai–Lidov oscillations (von Zeipel 1910; Kozai 1962; Lidov 1962) in the inner binary. This can in principle cause an enhancement in the eccentricity of the inner binary. In the case of EMRIs, however, we can expect gravitational perturbations in a hierarchical triple to be very weak. The pull of the third body essentially acts as a tidal force between the components of the inner binary. Therefore, it scales as ~aR⁻³, where a is the typical separation of the inner binary and R is the distance of the perturber to its centre of mass. For EMRIs, a will generally be very small (10 to 10³ Schwarzschild radii) and the third power of R will strongly suppress tidal forces. As an example, one can compare the von Zeipel–Kozai–Lidov oscillation timescale t<sub>KL</sub> with the gravitational radiation reaction timescale t<sub>GW</sub>:

$$t_{GW} \sim t_{KL} = 2\pi \frac{\sqrt{GM} R^3}{Gm_3 a^{3/2}} \quad (31)$$

By using Peters' formula (Peters 1964a) one can find the typical orbital separation at which GW emission and von Zeipel–Kozai–Lidov oscillations change the orbital elements on the same timescale. For circular orbits, this yields

$$R_{\text{KL/GW}} \approx a \left(\frac{m_3}{qM}\right)^{1/3} \left(\frac{a}{r_S}\right)^{5/6}, \quad (32)$$

where q is the mass ratio and M the total mass of the inner binary, while r<sub>S</sub> is the Schwarzschild radius of the central MBH and m<sub>3</sub> the mass of the perturber. For EMRIs, it is clear that the two timescales can be comparable only for very massive or very close perturbers. Nonetheless, recent works have used this result to compute event rates for binaries that are affected by the Kozai-Lidov eccentricity enhancement (Randall and Xianyu 2019b; Deme et al. 2020a), and Yunes et al. (2011b) showed
that a sufficiently large perturber (~10⁶ M☉) at sub-parsec distances can dephase the GW signal of an EMRI by a detectable amount.

The astrophysical community has mostly focused on understanding and modelling the influence of the environment on vacuum sources. Claims of detectability for any given effect are therefore often based on simple phenomenological criteria rather than complete signal injections and parameter inference methods. The simplest and most ubiquitous criterion is based on the concept of the SNR of a deviation from a vacuum waveform. While the SNR of a GW event can be estimated by the following formula:

$$ \text{SNR} = \sqrt{2 \cdot 4 \int_{f_{\text{min}}}^{f_{\text{max}}} \frac{df'}{S_t(f')f'^2} \frac{h_c^2(f')}{S_t(f')f'^2}} \quad (33) $$

The SNR of a deviation can be found by replacing $h_c$ with:

$$ h_c \exp(i\phi) \rightarrow h_c \exp(i\phi) - h_c' \exp(i\phi'), \quad (34) $$

where the primes denote the waveform of a source that is modified through the action of some environmental effect. For GW sources in general, an SNR value of ~8 is chosen as a threshold required in order to claim detectability. The same is assumed to be true for a given deviation, $\delta h_c$ which is deemed detectable whenever $\Delta \text{SNR} > 20$ (at least for EMRIs). While such criteria can serve as a first order approximation, they do not take into account many of the complications that will plague data analysis procedures required to extract signal from LISA's datastream. Effects such as degeneracies, the lack of appropriate waveforms and subtleties of Bayesian analysis in parameter spaces with many dimensions are only a few of the many considerations that should in principle be taken into account when considering the detectability of environmental effects. Degeneracies are especially important, since the influence of many environmental effects might be misinterpreted as sources with different intrinsic parameters. As a simple example, consider the evolution of a BH binary in gas. The primary effect of gas would be to change the rate at which the binary chirps, i.e. $f$. However, in a blind waveform template search, the same variation in $f$ could likely be accounted for by a slightly modified mass of the system, inducing a bias in parameter estimation. With sufficient SNR, it will be likely possible to break such degeneracies from leading order parameters such as chirp mass and distance, using e. g. the information contained in higher derivatives of the frequency, $\ddot{f}$. Further work should confirm that the same can be said for subtler GR vacuum effects, mainly the spin components and the eccentricity of the source. A possible way forward would be an increased collaboration between the data analysis and astrophysics working groups: the former can provide more sophisticated phenomenological detectability criteria, while the latter could direct the data analysis efforts towards those effects that are expected to be relevant.

### 3.4.6.1 The contribution of LISA to our understanding of the host environment
The variety of gas torques suggests that, if chirping, EMRIs in gaseous environments will exhibit characteristic signatures that may allow us to probe the inner regions of AGN discs. If a phase shift is detected and confirmed to be of gas
origin, its magnitude and evolution will be a direct consequence of interaction with local gas properties around the BHs. However, detecting such deviations will require improving models of environmental effects such that we can include them properly in parameter estimation. A critical step is to assess for which regions of parameter space these effects will be unique or degenerate with system parameters. In the latter case, neglecting them may induce biases in parameter estimation. We expect such signatures to arise in only a subset of systems, whereas deviations from general relativity would arise in all EMRIs (depending on the observed frequency). Gas effects can also lead to additional waveform implications: for example prograde, disc-embedded sources will likely have low eccentricity and some degree of spin/ inclination alignment with the central MBH. If deep within the potential well of the MBH, these effects may be complemented by a phase-shifting from the Doppler effect (Sect. 3.4.3).

Elaborating on what we have presented here but also in the previous sections, the presence of an XMRI can alter the evolution of an EMRI on its way to cross the event horizon of the MBH. This can lead to extreme situations in which the orbital dynamics of the EMRI is not just affected by the presence of the XMRI, but to the point of scattering off the EMRI from its inspiraling orbit towards the MBH (Vretinis and Amaro Seoane, in prep.). Since our Galactic Centre and MBH might be envisaged as a typical target for LISA, this means that many, if not all nuclei in the LISA observational volume are prone to this problem.

Our theory of how stars distribute around MBHs is more than four decades old and seems to be robust. However, at distances very close to the MBH, the power-law distribution of the stellar system (the “cusp”) is ill-defined, because the number density drops significantly. If a star happened to be close to an EMRI, in principle one could reverse-engineer the modulation induced in the waveform, in particular in the phase, to recover information about such perturbing stars from a region which is too obscured to be accessible to EM telescopes.

The merger in a b-EMRI system induces a kick to the BH remnant (Centrella et al. 2010). This kick causes a glitch in the EMRI waveform, which, through a careful analysis, is discernible in the data stream (Han and Chen 2019). The b-EMRI can hence accurately weigh the mass loss due to the BH+BH merger, and offer an opportunity to test general-relativistic effects (in particular the dispersion relation of GWs and the weak equivalence principle).

Since a significant fraction of EMRIs can be hosted in galaxies which move relative to us at very high speeds, the imprint of the aberration and beaming effects on the waveform can be crucial. A constant drift of the centre of mass of a source also can affect the higher multipoles of the gravitational waveform. This, in turn, affects the frequency and amplitude of the wave as seen by a distant observer (Gualtieri et al. 2008; Torres-Orjuela et al. 2021b). Therefore, higher modes can be used to break the aforementioned degeneracy between a constant velocity and the mass/distance of the source. Considering the change of the modes for EMRIs, LISA should be able to detect constant velocities of just 1000 km s⁻¹ for an SNR of around 70 (Torres-Orjuela et al. 2021a). Moreover, as mentioned in that work, we could use
this information to obtain a detailed map of the relative speed distribution of galactic clusters out to distances inaccessible to EM observations.

Another interesting possibility is that dark matter minispikes could impact the gravitational waveform, inducing dephasings that could be detected by LISA (Eda et al. 2013; Yue and Han 2018; Hannuksela et al. 2020; Kavanagh et al. 2020). Furthermore, the existence of dark matter halos around IMBHs could accelerate the formation of IMRIs (Yue and Han 2018). Therefore, the event rates of IMRIs may be much higher than previous estimates, which did not include a DM halo (Yue et al. 2019).

As for astrophysical chaos, we can deduce that the detectability of the plateau scales with the magnitude of the non-Kerr parameters and the mass ratio of the EMRI: the “islands” presented above become larger for greater non-Kerr parameters, and the system requires more time to cross an island for greater mass disparity (Lukes-Gerakopoulos et al. 2010). Crossing into an island leads to a period of frequency modulation, which can, in principle, be detected by LISA (Destounis et al. 2020). More careful data analysis (using, e.g., a Fisher matrix study) is therefore required to determine whether chaotic phenomena have a non-general-relativistic origin, since the frequency jumps described in Fig. 34 may be mimicked by environmental effects.

The research carried out so far demonstrates the difficulty of distinguishing gas-driven environmental effects from poorly modelled GR effects. Work to date has explored a relatively narrow range of parameter space for possible environmental effects, and more work should be done to understand the dominant effects, even within currently available 2D Newtonian hydrodynamical models. As computational resources increase, closer to launch, it would also be helpful to expand theoretical efforts to include at least some 3D general relativistic magnetohydrodynamical simulations. Though challenging, further work must also be done to examine the degeneracies between, e.g., additional PN terms and gas-driven departures from GR. The most important efforts are finding effects that share the same frequency or mass dependence—for effects that do not share dependencies, we can hope to distinguish the source from the GW observations themselves. Gas-driven sources should be a subset of all sources (and should have eccentricities and inclinations which help to distinguish them), while higher-order corrections should apply to every system (though the corrections may be mass- or frequency-dependent). Assuming we overcome the challenges listed here, disentangling various environmental signatures from GWs will give us access to unique measurements of MBH environments purely through GWs. Such measurements are inaccessible via EM observations. Characteristic deviations (or even a lack thereof) will provide constraints on gas densities, dark matter profiles, or the presence of external perturbers.
## 3.5 EMRI background

**Coordinators:** Pau Amaro-Seoane, Andrea Derdzinski

**Contributors:** Pau Amaro-Seoane, Andrea Derdzinski, Giuseppe Lodato, Martina Toscani

While the majority of this chapter describes resolvable sources (as they are certainly the most interesting for guaranteed science), most EMRIs/IMRIs throughout the Universe will not be individually detectable, particularly if they are too distant, at earlier stages of their inspiral, or their GWs are too weak (which is more so an issue for inspirals of WDs or NSs). The combined signal from the population of faint, unresolved sources will constitute a stochastic background.

The EMRI background may lie well below the LISA sensitivity or exceed it, contributing an additional confusion noise. Its amplitude scales with the EMRI rate, although not necessarily linearly (Barack and Cutler 2004a), and its precise spectral shape will depend on the efficiency of various formation channels over cosmic time. Seminal predictions find that the background signal will only become comparable to the LISA noise if the EMRI rate is substantial: e.g., if the detection rate is as high as $\mathcal{O}(10^2)$ detections per year, the corresponding background may increase the LISA noise by a factor of nearly $\sim 2$ (Barack and Cutler 2004a). More recent estimates based on EMRI catalogues by Babak et al. (2017) use an updated version of the LISA sensitivity curve and find that, for a range of EMRI rates, the background may add considerable noise (attaining an SNR of a few to few-hundred) within the LISA sensitivity bucket around $f \sim 3$ mHz (Bonetti and Sesana 2020). Higher levels of confusion noise may compromise our ability to detect faint sources that fall into this frequency range, such as high redshift, low mass MBH mergers–although one could argue that this would be compensated for by the generous resolvable EMRI rate.

A detectable background may provide additional information on the cosmological EMRI population and the efficiency of various formation mechanisms, if there exists robust differences in the spectrum between formation channels. The trick to detecting (and then hopefully characterizing) a background signal is to distinguish it from the instrumental noise as well as other confusion sources (see Romano and Cornish 2017 for a comprehensive review). For LISA, the main contributor to confusion noise is expected to arise from Galactic binaries: while many will be individually resolvable, the rest will form a confusion foreground (e.g., Nissanke et al. 2012, but see Chapter 1) that may overwhelm any extragalactic stochastic signal. Fortunately, LISA's orbital motion around the Sun introduces an annual modulation in the anisotropic galactic foreground, and this makes it possible to distinguish the astrophysical signal from the instrument noise. With a prior understanding of the LISA noise, knowing the distinct spectral shape of an astrophysical foreground further helps us separate the two, so there is hope for detecting an underlying stochastic background (Adams and Cornish 2014). Such techniques were successfully applied in the LISA mock data challenge (Robinson et al. 2008). At the moment, predictions for the EMRI background signal suffer from the same
<table><caption>**Table 9** Rates and SNRs for inspirals. Note that the rates for XMRIs are at any given moment in the MW and, possibly, nearby galaxies (see reference)</caption><thead><tr><th>Inspiral type</th><th>Rate (year<sup>-1</sup>)</th><th>SNR</th><th>References</th></tr></thead><tbody><tr><td>EMRI</td><td>10–10<sup>3</sup></td><td>~100</td><td>Amaro-Seoane (2018b, 2021), Babak et al. (2017)</td></tr><tr><td>light IMRI</td><td>6–60</td><td>10–10<sup>3</sup></td><td>Arca Sedda et al. (2021a), Amaro-Seoane (2021)</td></tr><tr><td>heavy IMRI</td><td>2–20</td><td>10–100</td><td>Miller (2005), Arca-Sedda and Gualandris (2018)<br>Arca-Sedda and Capuzzo-Dolcetta (2019)</td></tr><tr><td>XMRI</td><td>~ few tens (at any given moment)</td><td>10–10<sup>4</sup></td><td>Amaro-Seoane (2019, 2021)</td></tr></tbody></table>

uncertainties as detection rates (see Table 9), but these can be improved as we increase our understanding of formation mechanisms. Improving waveform modelling or finding other methods of accurate signal extraction will also be critical if we hope to detect an underlying signal.

An important step in this analysis would be to distinguish the EMRI background from other possible background sources. In addition to EMRIs, there may be characteristic background signals from extra-galactic WD, NS, or BH binaries (e.g., Chen et al. 2019b; D’Orazio and Samsing 2018), TDEs (discussed below), phase transitions in the early universe (Maggiore 2000; Giblin et al. 2012; Leitao et al. 2012), or cosmic strings (Siemens et al. 2007). If the spectrum is sufficiently constrained, then it is likely that the origin of the signal, whether from a large number of unresolved EMRIs or other extragalactic sources, can be determined (Barausse et al. 2020a).

### 3.5.1 TDE background

A particular type of EMRI background is the one generated by the unresolvale GW signal from the cosmic TDE population. The calculation of such background has been performed by Toscani et al. (2020) for both main-sequence stars being disrupted by MBHs in galactic nuclei and WDs being disrupted by IMBHs in globular clusters. The signal has a characteristic spectral shape $h_c \propto f^{-1/2}$, due to the specific impulsive nature of these events. The predicted amplitude of the background is generally low, with WDs on IMBHs providing typical strains of $\approx 10^{-23}-10^{-21}$ and main-sequence stars on MBHs providing $\approx 10^{-22}$.

#### 3.5.1.1 The contribution of LISA to our understanding of backgrounds of inspirals
In summary, LISA will have the capability to detect a stochastic background signal, once the galactic foreground is subtracted. This measurement will improve throughout the mission lifetime as we constrain the instrument noise
and resolve individual sources (Adams and Cornish 2014). If EMRIs provide the dominant contribution within some frequency range—e.g., around 3 mHz, as predicted by Bonetti and Sesana (2020)—a measurement of the background spectrum can serve as an additional measurement of dynamics in galactic nuclei. For example, the amplitude and spectrum of the background are related to the number of EMRIs that are either at earlier inspiral stages or at higher redshifts, as well as the MBH mass function. If the background is above the LISA sensitivity and not well-characterised, it will contribute to the noise budget, possibly complicating the detection of other weak signals. To avoid this, it is important that we improve our predictions on the EMRI rate. Further work is also needed to constrain the expected spectrum of various background signals, in order to determine which will be dominant, distinguishable, and removable from the LISA data. If the EMRI rate is low enough such that the background falls below the LISA sensitivity, then it becomes possible to detect other stochastic signals, such as those predicted from extragalactic binaries or signatures from the early Universe.

As for the background from TDEs, its detection could provide interesting insights both on the distribution of quiescent MBHs (for main sequence stars tidally disrupted) and on the occupation fraction of IMBHs in globular clusters (for TDEs of WDs), up to redshift $\approx 3$. Yet, this detection seems to be very difficult. Indeed, the background produced by WD TDEs will lie in a high part of the frequency window (deciHertz to a few Hertz), where LISA will be less sensitive (yet, more sensitive foremeters in this frequency interval are planned for the future). Instead, the background from MS TDEs is expected at lower frequency ($10^{-4}-10^{-2}$ Hz), but will be still below the threshold LISA sensitivity. Hence, this detection seems unlikely (although some background signal below the strain sensitivity might still be visible in some cases; see Sesana 2016 for more details).

## 3.6 Conclusions

### Contributors: Saavik Ford

To summarise: the basic physics of EMRI mergers has been known for a long time. We can expect to find EMRIs in NSCs harboring an MBH, and can predict the dynamics of their formation and evolution using relaxation theory. The waveform modelling for EMRIs is also reasonably advanced, such that the path to detectability of such signals is understood (if challenging). However, there are substantial uncertainties in the astrophysical parameters that govern the rates of EMRI production, notably the spins of MBHs and, most critically, the radial mass distribution of NSCs. These astrophysical unknowns will change the ratio of plunging extreme mass ratio mergers to bona fide EMRIs—in the case of plunging mergers, an extreme mass ratio merger does occur, but the interactions of the merging object with the inner edge of the stellar cusp alters the trajectory of the low mass object after only a few orbits, and produce a rapid merger. Since LISA detection of EMRIs will depend on the buildup of a sufficient SNR over many orbits, plunges are rendered undetectable.
However, assuming the inner edge of an NSC is typically sufficiently far from the MBH, and if MBH spins are typically non-zero, EMRIs can occur at a high enough rate that LISA would detect one or more over the lifetime of the mission. If an EMRI is detected, we will immediately obtain a wide variety of both fundamental physical and astrophysical information (including, implicitly, information about the mass distribution in NSCs and MBH spins). Because of the many-orbit nature of an EMRI, such events can provide a detailed map of the gravitational field in the vicinity of the MBH, yielding exquisite measurements of the mass and spin of that MBH, and providing an opportunity to probe fundamental physics by testing for subtle departures from GR.

Given the current uncertainties on EMRI rates, it is most useful to proceed along multiple fronts:

- Theoretical work to understand higher-order dynamical effects which may preserve more EMRIs for sufficient cycles to allow detection by LISA (i.e. preventing plunges)

- Observational work constraining the inner edge of the NSC cusp in nuclei other than the MW (M32 would be notably useful)

- Theoretical and observational work constraining the binary fraction in typical NSCs (enabling better estimates of the EMRI rate due to binary tidal separation)

- Theoretical work to develop non-standard EMRI channels, especially AGN and SNe routes, to constrain rates and parameter distributions, and permit reverse engineering of astrophysical parameters

In addition, there is groundwork to be done on the waveform, data analysis and coordination front:

- Self-force calculations to second order in mass ratio for generic orbits in a Kerr metric to enable high precision waveform calculations

- Further data analysis work with updated waveforms to improve EMRI extraction from the LISA datastream

- Coordination of data analysis with radio and ground-based GW observatories in case of pulsar or b-EMRI detection

IMRIs provide still more exciting science opportunities, but correspondingly more challenging uncertainties. Due to their larger mass ratio, IMRIs cannot be treated using the same theoretical mechanisms as EMRIs (i.e. as small perturbations), yet they are also not sufficiently large to be treated using the mechanisms that apply to near-equal-mass binaries. This reality provokes difficulties in several directions—we cannot generally apply the same relaxation theory strategies to predict IMRI formation dynamics, nor can we readily produce IMRI waveform models using numerical relativity without changing computational strategy. In addition, there are at least 2 types of IMRIs to be considered: (1) light IMRIs, where the more massive partner is an IMBH and the less massive partner is a stellar mass BH; and (2) heavy IMRIs, where the more massive partner is an MBH and the less massive partner is an IMBH. There are multiple formation channels for each, and thus large astrophysical uncertainties in predicting their rates. One substantial uncertainty has recently been
removed: with the announcement of GW190521, we are certain that low-mass IMBHs do exist. Though their formation environment remains uncertain, work thus far points to some kind of dynamical origin, encouraging expectations that there may be environments conducive to the formation of at least some light IMRIs.

Broadly, the channels for light IMRIs include: (1) formation in globular clusters (assuming the presence of an IMBH in the cluster); (2) formation in dwarf galaxies (assuming the presence of an IMBH in the galaxy); (3) formation in an AGN disc (assuming the formation of an IMBH at a disc migration trap). In each of these cases, the first uncertainty is the presence of an IMBH in the relevant environment. There are no universally accepted detections of IMBH in globular clusters. While there are some IMBH known in dwarf galaxies, their occupation fraction is not well-measured and depends on the still unknown physics of BH seed formation at high redshift. IMBH formation in AGN discs is expected to be nearly universal if such discs contain a migration trap and are sufficiently long-lived. Unfortunately neither condition is sufficiently theoretically or observationally well-constrained to make a confident statement on the rate of IMBH formation in AGN discs.

Sadly that is not the end of our uncertainties for the formation of these systems—the dynamics in each case are difficult to model, as noted above, and as with EMRIs, the systems might produce a beautiful, detectable inspiral, or a rapid plunge. Among the important open questions are the observational presence or absence of IMBH in each formation environment, and observations and theoretical investigation of the (extremely different) dynamical environment around each IMBH.

For heavy IMRIs, we have a similar diversity of formation channels: (1) globular clusters containing an IMBH infalling into a galactic nucleus containing a MBH; (2) dwarf galaxies infalling into a galactic nucleus containing a MBH; (3) AGN-produced IMBH falling into their host MBH (likely in the post-AGN phase). Globular clusters, being denser than dwarf galaxies, will deposit their IMBHs in galactic nuclei more rapidly than dwarf galaxies, and in general, are expected to dominate the rate in this formation channel. However, if globulars do not contain IMBHs, we should consider dwarf galaxy mergers quite carefully, since at least some dwarfs are known to harbor IMBHs. For low-mass galaxy groups, dwarf mergers will be the most common type of merger and do lead to the formation of bound IMBH-MBH systems in less than a Hubble time. However, theoretical astrophysical rate calculations for resulting IMRIs, over the volume probed by LISA, remain an important open question. Observations of dwarf galaxies and their evolution over cosmic time, as well as observations that inform the occupation fraction of globular clusters and dwarf galaxies are, consequently, critical unknowns. AGN production of IMBHs provide a potentially extremely high rate of IMRIs, given the formation location and expected GW inspiral time; such IMRI systems will not likely be disrupted by interactions with the NSC. However, as with the light IMRI channel, substantial uncertainties in the structure of AGN discs and their lifetimes lead to orders-of-magnitude uncertainties in the rate estimates for this channel.

IMRIs have received less attention in the literature to date, and consequently the tasks to complete before LISA launches tend to be larger. These include:

- Determining the occupation fraction of globular clusters and dwarf galaxies
- Determining the contribution to the heavy IMRI rate from low mass galactic environments

- Determining the formation environment of GW190521-like sources

- IMRI waveform modelling and extraction (may require expensive numerical relativity modelling)

- Modelling of IMRI formation (and rates) in both gas-poor and gas-rich environments

Finally, we consider the open questions related to the relatively new class of unequal mass ratio inspirals, XMRIs. Here, the physics of their formation and evolution is similar to that of EMRIs, but they are labelled eXtreme due to the very small mass of the secondary—for LISA frequencies, the secondary would typically be a brown dwarf. Similar to the situation for EMRIs, the uncertainties largely relate to our astrophysical ignorance; however, we are able to limit the locations we must investigate. Due to their low mass and consequently small strains, XMRIs will only be detectable within roughly our Local Group, meaning either from the MW or M31. In addition to the relevant questions for EMRIs (especially NSC radial mass distribution), we must understand the mass function of those NSCs. How many brown dwarfs are there in the galactic centre? If the IMF is top heavy, there may be very few—however, if we assume a more standard IMF, and the radial mass distribution and MBH spin are favorable, LISA can expect to detect one or more XMRIs from Sgr A* over the mission lifetime.

For XMRIs there are several useful items to work on as we proceed towards launch:

- Determine the low end mass function in the galactic centre

- Investigate possible interactions between XMRIs and EMRIs (and find distinguishing observables between interacting EMRI-XMRI systems and departures from GR such as the chaotic behaviour introduced by non-Kerr objects)

- Data analysis work to properly characterize potential loud XMRIs

As we have discussed, various mechanisms for producing EMRIs and IMRIs may have EM counterparts—this may enable independent rate constraints either prior to or concurrent with LISA; further, if specific counterparts are reliably identified, we can use the complementary information provided by each messenger to learn more about the astrophysics of the emitting system. Notable work to be done includes:

- Detailed hydrodynamical models of EM emission mechanisms for GW events in AGN discs

- Athena observations of candidate IMRI systems if Athena will not be flying simultaneously with LISA (or coordination if missions are concurrent)

Each of these types of inspirals present a large parameter-space of possible waveforms, making detection itself a notable challenge. Narrowing the theoretical uncertainties for each channel, in advance of LISA, therefore also has implications for the detectability of their signals. However, work may also be done on the signal processing side to find new strategies for extracting the signals, and doing reliable
parameter estimation on them, from the LISA data stream. In this context, there is
currently substantial concern over possible degeneracies between gas-induced phase
shifts and departures from general relativity; fortunately, if we have multiple events
from multiple populations, departures from general relativity should be universal,
while gas effects will impact only a subset of events. On the other hand, some signals
containing environmental effects will be clearly identifiable (e.g., b-EMRIs). All of
these channels may contribute to a detectable EMRI background; however,
disentangling multiple populations from such an unresolved background will be
extremely challenging and fundamentally requires more theoretical development
from each contributing channel. From these areas, we would especially like to
highlight the need for a thorough parameter space exploration with at least 3d
Newtonian hydrodynamical simulations of the impact of gas on the inspiral
waveforms.

Substantially unequal mass ratio inspirals of all types represent an important class of sources, uniquely detectable by LISA. In order to best exploit the astrophysical and fundamental physical science achievable by LISA using these types of events, in the years preceding launch, we will need to work primarily on developing more detailed models for each formation channel, and on observationally constraining the parameters used as inputs of those models, along the lines we have outlined above.

## 4 General summary

The decade prior to LISA's launch will be an exciting one for the astrophysics community, presenting unique challenges and opportunities in preparing for LISA's first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and current astronomical observations that will influence preparations for the pipelines that will deliver LISA data, and guide our interpretations of the first LISA observations and catalogues.

This review describes the current state of knowledge regarding three main source
classes for LISA: ultra-compact stellar-mass binaries, massive black hole binaries,
and extreme or intermediate mass ratio inspirals. For each of these three source
classes, our current understanding of the astrophysical processes that create them and
guide their ongoing evolution is a rich tapestry formed from extant observations
(usually electromagnetic), numerical simulations and modelling, and theoretical
considerations. LISA data will be added to this, providing new independent
information that will help constrain the physics governing these systems, and
opening up new avenues of investigation for future observations, theory, and
simulations.

Astronomy observations will continue to evolve and alter the scientific landscape prior to LISA's launch, and theory and modelling will become more refined. Such advances will inform our understanding of the ways in which LISA data can be used, and they can also sharpen the focus on the important ways in which gravitational wave data will expand and enhance our ability to understand astrophysical phenomena in many different environments and scales. This review endeavours to provide a framework within which to consider these possibilities, and should be a
good starting point for those interested in using LISA as a new observational tool for understanding the Universe.

**Acknowledgements** P. Dayal acknowledges support from the European Research council (ERC-717001) and from the Netherlands Research Council NWO (016.VIDI.189.162). P.H. Johansson acknowledges the support from the European Research Council (ERC-818930). S. Toonen acknowledges support from the Netherlands Research Council NWO (VENI 639.041.645 Grants) C. Unal is supported by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS - CZ.02.1.01/0.0/0.0/15_003/0000437). S. Chaty acknowledges the LabEx UnivEarthS for the funding of Interface project I10 "From binary evolution towards merging of compact objects". A. De Rosa acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.O E. Berti is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and NSF Grant PHY-20043. D. Gerosa is supported by European Union's H2020 ERC Starting Grant No. 945155-GWmining, Leverhulme Trust Grant No. RPG-2019-350 and Royal Society Grant No. RGS-R2-202004. T. Bogdanovic acknowledges support by the NASA award No. 80NSSC19K0319 and by the NSF award AST-1908042. D. Porquet acknowledges funding support from CNES. C. Danielski acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709) B.L. Davis acknowledges support from Tamkeen under the NYU Abu Dhabi Research Institute Grant CAP3. F. Pacucci acknowledges support from a Clay Fellowship by the SAO and from the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. A.J. Ruiter acknowledges support from the Australian Research Council Future Fellowship Grant FT170100243. V. Paschalidis is supported by NSF Grant PHY-1912619 and NASA Grant 80NSSC20K1542 to the University of Arizona, and NSF-XSEDE Grant TG-PHY190020. D. Haggard acknowledges support from the NSERC Discovery Grant and Canada Research Chairs programs, and the Bob Wares Science Innovation Prospectors Fund. M. Toscani acknowledges European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 823823 (RISE DUSTBUSTERS project) and COST Action CA16104 - Gravitational waves, black holes and fundamental physics, supported by COST (European Cooperation in Science and Technology). M. Chruslinska, A. Istrate and G. Nelemans acknowledge support from Netherlands Research Council NWO. T. Fragos and S. Bavera acknowledge support from a Swiss National Science Foundation Professorship Grant (project numbers PP00P2_176868 and PP00P2_211006).

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