<table><caption>**Table 6.** The table shows the number of SNe Ia per year. The first row gives the number per year of SN Ia based on the observations of ZTF and ATLAS. The second row gives the rate corrected by the effective sky areas of about 65% surveyed by ZTF and ATLAS.</caption><thead><tr><th rowspan="2"></th><th colspan="4">Maximum distance</th></tr><tr><th>10 Mpc</th><th>20 Mpc</th><th>50 Mpc</th><th>100 Mpc</th></tr></thead><tbody><tr><th>Observed rate</th><td>0.7 ± 0.3</td><td>1.8 ± 0.5</td><td>17.3 ± 1.7</td><td>86.7 ± 3.8</td></tr><tr><th>Corrected rate</th><td>1 ± 0.4</td><td>2.7 ± 0.7</td><td>26 ± 2.6</td><td>130 ± 5.7</td></tr></tbody></table>

number of spectroscopically confirmed SN Ia as a function of distance results to be 4 within 10 Mpc, 11 within 20 Mpc, 104 within 50 Mpc and 520 within 100 Mpc resulting in the SN Ia rates per year given in table 6 (first row). These numbers have been corrected by a factor of about 1.5 considering the average coverage of 65% of the sky by the above surveys (second row of the table).

The numbers thus estimated are conservative. A non-negligible fraction of SNe are not classified and could potentially increase the SN Ia number, and detection incompleteness can start to arise at large distance due to the limited surveys' sensitivity. However, both biases can be considered negligible within 20 Mpc.

Considering a merging frequency of 0.1 Hz for a $1+1 M_{\odot}$ DWD system and the LGWA DWD horizon distance that continuously increases with observation time up to 10 yr, we estimate a total number of about 20 DWD mergers producing SNe Ia possibly detectable by LGWA over the 10 years of mission lifetime. This number does not take into account that some SN Ia are not detected by ZTF/ATLAS due to obstruction of the EM radiation, but such systems would not escape detection in GWs.

These detections enable the interesting possibility of measuring DWD systems in galaxies that in the past hosted a SN Ia. In that case, host galaxy distances derived from the GW signal might be used as an independent probe to calibrate the SN distances for cosmology. It should also be noted that DWD mergers cannot produce a SN Ia when the total mass of the binary lies below the Chandrasekhar limit. This means that many more DWD mergers should be expected to be observable with LGWA than the number inferred from SN Ia observations, and those could be used equally well to calibrate SN distances.

**Massive black hole binaries and extreme/intermediate mass ratio inspirals.** Massive BHs inhabit the nucleus of many galaxies [513, 514], including our own Milky Way, and 'feedback' from AGN is one of the most important processes that determine the evolution of galaxies. Understanding the formation of such heavy objects remains the subject of open debate [515]. Still, several channels have been proposed in the literature: objects formed with initially up to $\sim 10^3 M_{\odot}$ (the so-called *light seeds*) can be the remnants of Population III stars [516, 517]; otherwise, rarer and heavier objects ($10^4-10^5 M_{\odot}$, referred to as *heavy seeds*) can form through the direct collapse of pristine gas possibly going through an intermediate phase of *supermassive star* that eventually collapses into a MBH [518, 519]. Furthermore, MBHs can be generated through runaway collisions of young stars formed into a very dense star cluster [520] or through repeated mergers of smaller stellar black holes again within very
dense stellar environments [521, 522]. The latter scenario is of particular interest for the LGWA, as the GW signal produced the repeated inspirals of black holes can be probed by future GW missions [523]; (this aspect is further covered below when addressing EMRI/IMRI sources). After their formation, these objects are expected to grow through gas accretion, accretion of stellar objects and mergers with other MBHs to reach their present-day masses, correlating with the host galaxy masses ($M_{BH}-\sigma$ relation, [513, 514]). The GW detection of these sources across a broad range of masses and redshifts would thus provide unprecedented constraints on their still debated origin and growth across cosmic time.

*Massive Black hole binaries with nearly equal masses.* MBH binaries can form as a result of galaxy-galaxy mergers, a very common process according to the hierarchical clustering paradigm predicted by the Λ-CDM cosmological model. Given many galaxies are observed to host quasars since the very early Universe, it is reasonable to expect that many MBH pairs have found themselves in the centre of the same galaxy merger remnant across the history of our Universe. If the separation between the MBH pair shrinks down to separations ≪ 1 pc, GWs drive their final coalescence [524]. The GWs released in the process can be observed by current and upcoming GW detectors. In particular, LGWA will be mainly sensitive to merging MBHs in the mass range $10^3–10^6 M_\odot$. The detections of GWs emitted from these sources can provide robust constraints on the MBH merger rate and thus unprecedented information on the formation and evolution of MBHs over cosmic time. Additionally, combining GW detections and EM observations will provide insights on the origin of the $M_{BH}-\sigma$ relation.

At the early stage of the galaxy merger, the two MBHs are initially separated by hundreds of kpc. A series of mechanisms are required to ensure their sinking to the GW driven regime. Initially, the dynamical friction against dark matter, stars and gas induce the orbital decay [525], even if large-scale torquing sources such as bars, spirals, massive clumps and other irregularities may randomize the inspiral timescale [526, 527]. If the mass of the two MBHs (and merging galaxies) is not too dissimilar ($q = m_2/m_1 \sim 0.1–1$) the inspiral is likely to be completed within a timescale of the order of a Gyr, and the two MBHs find themselves bound in a pair. At this point, repeated three-body interactions with stars passing close to the binary, possibly combined with the presence of a gaseous circumbinary disk, are expected to sustain the shrinking all the way down to the GW driven regime in a timescale of a fraction of a Gyr [528], although recent numerical studies found that the physics of gaseous dynamics around the MBH can have a non-trivial impact on the binary evolution (see e.g. [529–532]). Note that, especially when the bound binary evolution is inefficient (e.g. if the properties of the gaseous disc do not induce negative torques on the binary, or the surrounding stellar population is scarce), another MBH can be brought onto the evolving binary by a further galaxy merger, inducing a triple interaction [533]. The dynamical evolution of the triple system often results in the merger of at least two of the three involved MBHs [533].

Several studies have assessed the merger rates of MBH binaries as a function of the mass of the merging bodies and redshift [101, 534] employing semi-analytical models for the cosmic evolution of galaxies and MBHs. The obtained predictions can vary substantially depending on the selected black hole seed mass (light versus heavy seeds) and physical prescriptions implemented to describe the physics of the inspiral; the intrinsic number of MBH mergers is typically of order a few tens to a few hundreds per year, with light-seed models predicting a
larger number of events dominated by mergers of MBHs with $< 10^4 M_\odot$, while the heavy seed models tend to feature fewer events, mainly involving MBHs of $10^4 - 10^6 M_\odot$. The LISA GW detector (which is sensitive to larger masses and higher redshift, if compared with LGWA) will likely have its detection rates (again of tens to hundreds of events per year) dominated by mergers occurring at $z \sim 5$ [101]; this suggests that the LGWA will be able to detect at least a few (tens) of events, and possibly have its detection rates dominated by lower-$z$ mergers bridging the gap. Possible observations of MBH mergers with LGWA in the mass range $< 10^4 M_\odot$ would bring new insights about the rate of low-mass galaxy mergers and the possible collision of massive star clusters (see also figure 5).

*Extreme and intermediate mass-ratio inspirals.* Binaries that have largely disparate mass ratios are referred to as EMRIs or as IMRIs. In the literature, EMRIs are generally defined as sources with mass ratio $q < 10^{-4}$, while IMRIs identify binaries having $q = 10^{-4} - 10^{-2}$. Given that LGWA would be sensitive to E/IMRIs occurring around MBHs with $\sim 10^2 - 10^5$, according to figure 5, many sources involving an MBH and a stellar compact object (WD, NS, or stellar BH) would fall into either of these categories.

As shown in section 2.2.1, LGWA is suited to detect binary mergers with component masses below $\sim 10^5 M_\odot$, generally referred to as IMBHs. The current debate about the existence and origin of IMBHs can be summarized with a question: are IMBHs the link between stellar BHs and SMBHs, or do they require a different formation channel from SMBHs? If IMBHs are the precursor of SMBHs, they must form from the first stars. Little is known about their origin as well. If these stars can attain masses above $10^3 M_\odot$, the so-called supermassive stars (SMS), they likely undergo direct collapse owing to general relativistic instability (SMSs) (see [535] for a review). However, recent numerical simulations predict that these ancient stars form with smaller masses, although this depends on the background ionization and accretion rate (see [536] and reference therein). Alternatively, SMSs can form through runaway collisions in star clusters, provided that its mass overcome the range affected by the PISN mechanism. The IMBH seed formed from the collapse of the SMS can further grow via accretion of stellar material or by merging with other BHs. Note that, however, the amount of matter accreted onto a BH during a star-BH interaction is unknown, and gravitational recoil prompted by asymmetric GW emission during BH mergers can eject the BH remnant from the cluster and hamper the development of further hierarchical mergers. Therefore it is important to obtain observational constraints for a wide range of MBHs, not only for stellar astrophysics but also for galaxy evolution.

This gives the mission a tantalizing advantage over milliHz detectors for probing lower mass systems such as the nuclei of dwarf galaxies or IMBHs in star clusters, for which observational evidence is severely limited (for recent reviews see [514, 537]). The few observational constraints about these elusive objects come from the detection of IMBHs in dwarf galaxies (see e.g. [538, 539]), the recent observation of GWs from merging BHs whose final remnants have masses $< 200 M_\odot$, like GW190521 [369], as well as systems for which a stellar tidal disruption occurred, which temporarily illuminates the MBH's environment [514, 540–543]. Further observations of IMBH candidates in the $10^3 - 10^4 M_\odot$ mass range have been observed in galactic and extragalactic globular clusters, but all such observations are controversial and widely debated [544–559]. As seen in figure 5, detection
of an IMRI merger with a primary of $10^5 M_{\odot}$ is possible, albeit at relatively low redshift ($z \lesssim 10^{-1}$).

The formation of binaries containing IMBHs may occur via several pathways, such as within dense stellar environments like dense young massive, globular, or nuclear clusters, or in the accretion disc of active galactic nuclei (see e.g. [404, 521, 522, 560–564]). These events may be a result of

(i) repeated collisions among massive stars occurring in the core of star clusters that build up a very massive star which ultimately collapses to an IMBH;

(ii) accretion of stellar material onto stellar-size BHs;

(iii) multiple stellar BH mergers (hierarchical mergers).

In the bright AGN systems for which a radiatively efficient accretion model is invoked to explain the emission, the gas configuration is expected to be dense enough to influence the orbits of surrounding stars and BHs [408, 565, 566]. Such disks may also become vulnerable to fragmentation in certain conditions, in which case they form stars and subsequent BHs in situ [567–570]. Accretion disks in AGN are considered potential factories for IMBH mergers and light IMRIs given that the gas-rich environment can facilitate the interaction between BHs, IMBHs, and MBHs, in addition to providing them material to accrete and grow. In these environments the formation mechanisms of each of these events are intimately connected, and LGWA will fill in a piece of the puzzle with detections of IMBH interaction. This will be complementary to milliHz space-based detectors, which will be sensitive to more extreme mass ratio events (EMRIs), while potentially missing out on the lower mass binaries which could be the building blocks (see e.g. [562, 571]).

Recent analysis of the latest LVK detections suggest that at least a fraction of detected BH-BH mergers may occur within dense stellar environments or AGN (e.g., [572], although see [573]). The motivation for this is to explain events with particularly high masses, residual evidence of eccentricity [436, 574], spin alignment vs. antialignment [575] and events which have associated EM counterparts [576, 577], as variations of each of these may be a result of either dynamically-driven or gas-driven encounters. The AGN-assisted formation channels may skew towards high mass ratio mergers [578], strengthening the case for IMRI production in these environments. LGWA will provide valuable information on the earlier stages of these systems, as well as new insight into the more massive systems which may be natural by-products of hierarchical and gas-assisted evolutionary pathways. The detection of IMRIs involving IMBHs by LGWA could thus help to shed light on the formation channel of these elusive objects. In particular, the IMBH formation, and related probability, intrinsically depends on the IMBH formation channels discussed above.

So far, we have discussed EMRIs and “light IMRIs” involving a stellar object as a secondary. Another class of IMRIs is often identified, involving an IMBH and a SMBH residing in a galactic nucleus [579–584]. Several formation scenarios can contribute to the formation of these heavy IMRIs, like galaxy mergers, star cluster orbital segregation, and migration in AGN discs. The first channel assumes that an IMBH-SMBH binary can form in consequence of a minor merger. In the second scenario, massive and dense star clusters
forming in the inner region of the galaxy nurture the formation of an IMBH while being
dragged by dynamical friction toward the galaxy centre. Eventually, the infalling cluster
disrupts and its IMBH librates in the galactic nucleus, further spiraling in via dynamical
friction as well and finally pairing with the galactic SMBH. In the third scenario, an IMBH
forms in an AGN disc, possibly via a combination of processes like migration trap dynamics
and repeated mergers (see previous paragraphs) and slowly migrates inward via gaseous
dynamical friction, eventually binding to the SMBH.

To build a comprehensive picture of MBH, IBMH, and BH interactions across the mass
spectrum, LGWA presents a promising case to bridge the gap between the SMBH sources
detectable in the milliHz regime (with LISA or TianQin) and high frequency sources that will
be detected with increasing sensitivity with next-generation terrestrial detectors. In this gap
lie the early inspirals of stellar-origin BH binaries, the mergers of IMBHs, and the potential
interplay between the two (IMRIs), all of which may form in dense stellar environments and
in galactic nuclei. Each scenario has peculiarities in terms of occurrence frequency or IMBH
mass spectrum, thus the future detection of GWs emitted by both light and heavy IMRIs
will represent a key to unravelling the true nature of IMBHs.

**Tidal disruption events.** As discussed in section 3.2.1, the predominant and ubiquitous generation mechanism for TDEs is believed to be two-body relaxation: stars in the vicinity of the central massive black hole tug each other's orbit, so that the trajectory of an unlucky star may eventually attain a very small pericentre and get disrupted (e.g. [585]). Relying on the so-called *loss-cone theory*, theoretical models predict typical TDE rates between $10^{-6} - 10^{-3}$ events per galaxy per year [586, 587].

The accumulation of TDE observations over the last few years has made it possible to
compare those predicted rates with the observed ones and even estimating rates as a function
of the black hole and galaxy mass [588, 589], with reasonable agreement with theoretical
models (at least at the order-of-magnitude level). A peculiar aspect of the host galaxies of
TDEs is the fact that post-starburst systems (which underwent a substantial episode of star
formation in the last ~1 Gyr) appear to be overrepresented by a factor of a few tens [590, 591].

LGWA is going to predominantly observe TDEs occurring about relatively low-mass MBHs
(< $10^6 M_\odot$), likely inhabiting dwarf galaxies or globular clusters in the local Universe [514].
This poses a series of challenges for the proper modelling of the rates, as the occupation fraction
of MBHs in those low-mass systems remains debated [592], as well as their innermost stellar
density profile (which remains virtually unresolvable at the scale of a fraction of the MBH in-
fluence radius [593], where most of TDEs are expected to be generated [594]). Although earlier
works have suggested TDE rates would be larger about low mass MBHs [586, 595], this idea has
been recently revised and it seems instead that rates about low-mass MBHs drop significantly
over time after an initial burst [594, 596]. Overall, the event rates about low-mass MBHs
in the local universe remain under close investigation, and upcoming surveys as LSST/Vera
Rubin Observatory as well as ULTRASAT will greatly improve our understanding of this in
the next few years, thus allowing to refine the science we expect to obtain from the LGWA.

**Gravitational waves from asymmetrical neutron stars.** Rotating neutron stars emit
continuous GWs if they present some sort of non-axisymmetry [597, 598]. The simplest form
<figure><img src="image_32.png" /><figcaption>**Figure 29.** Lines: smallest detectable signal from known pulsars assuming 1 year of observation time. Circles: spin-down upper limits for known pulsars. The left plot shows the intrinsic GW amplitude, whereas the right plot the ellipticity associated with the spin-down limits.</figcaption></figure>

of this is an equatorial ellipticity $\epsilon$, defined as $\epsilon = (I_{xx} - I_{yy})/I_{zz}$ with $I_{ij}$ being the moment of inertia tensor of the star and the rotation axis being in the $z$ direction. The GW signal is nearly monochromatic at a frequency $f$ equal to twice the rotation frequency of the star, and its amplitude $h_0$ at distance $d$ from the source is

$$h_0 = \frac{4\pi^2 G \epsilon I_{zz} f^2}{c^4} \quad (3.7)$$

For neutron stars observed as pulsars, the frequency and frequency derivatives are measured and the GW amplitude $h_0^{\text{spdown}}$ that could account for the observed kinetic energy loss (i.e for the observed spin-down) can be calculated as

$$h_0^{\text{spdown}} = \frac{1}{d} \sqrt{\frac{5GI}{2c^3} \frac{|\dot{f}|}{f}} \quad (3.8)$$

The spin-down amplitude is the maximum GW signal amplitude that one can reasonably expect from a star rotating at frequency $f$ and with a spindown $\dot{f}$, and it is an important benchmark because if the sensitivity of a search does not at least reach this level, it could be argued that such search does not stand a chance of detecting a signal.

Searches for emissions by pulsars with positions and frequency evolution known from radio observations are the most sensitive among all GW searches. In order to gauge the performance of LGWA to continuous GWs, this is a good place to start. The left panel of figure 29 shows that the bulk of the known pulsars emitting GWs in the LGWA frequency band are emitting below the smallest signal detectable by LGWA. The latter is estimated by assuming a search over 1 year of data with a detector having the power spectral density of the Si configuration divided by 4 (see section 2.1.5). Furthermore, the spin-down ellipticities of the known pulsars rotating with frequency above 0.05 Hz (GW frequency 0.1 Hz) are all greater than $10^{-4}$, as shown in the lower panel of figure 29. Since the maximum ellipticity that a neutron star crust can sustain before breaking is likely not greater than $10^{-5}$ [599–602], emission at the spin-down level for these objects appears somewhat unlikely setting stronger requirements on the sensitivity to observe their GW emissions.
Assuming a factor of ≈ 2 better sensitivity, corresponding to searching about a decade of data corresponding to the targeted lifetime of LGWA, the number of detectable pulsars — i.e., with spin-down amplitude larger than the expected sensitivity level — grows to about 10, but all with very large ellipticities.

The low frequency range is very challenging for the detection of continuous GWs because, for a given physical scenario, say a given ellipticity value, the GW amplitude scales with a fairly high power of the frequency ( ∝ f² for deformations, from eq. (3.7), and ∝ f³ for r-mode emission), making the signal much weaker than at higher frequencies. Whereas signals at low frequencies are emitted by neutron stars with a broader range of magnetic field values than those emitting in the band of terrestrial detectors [603], the prospects for detection would significantly increase with a sensitivity ≈ 20 times larger. Relative to the results shown in figure 29, another factor of 3 in sensitivity can be achieved by considering a 10-year observation time with LGWA corresponding to its targeted mission lifetime, but observation of GW emissions from spinning neutron stars remains unlikely.

Massive magnetars are young and highly magnetized neutron stars with an unknown origin. The newly formed massive magnetars spin down mostly through magnetically induced GW emission and magnetic dipole radiation. The evolution equation for a magnetar’s angular frequency can be written as

$$ \dot{f} = -\frac{2\pi^2 B_d^2 R^6 f^3}{3Ic^3} - \frac{512\pi^4 G\epsilon^2 I f^5}{5c^5}. \qquad (3.9) $$

The dipole magnetic field of the magnetar at the pole is $B_d$. Different interior magnetic field configurations, such as poloidal-dominated, toroidal-dominated, and twisted-torus configurations, indicate various relationships between the magnetic field and ellipticity [604]. Detectors will probe various configured models, and they will be able to explore the parameter space in these models.

### 3.2.3 GW cosmology

*Main contributors:* Francesco Iacovelli, Enis Belgacem, Marica Branchesi, Stefano Foffa, Arun Kenath, Michele Maggiore, Michele Mancarella, Suvodip Mukherjee, Niccolò Muttoni, Masroor C. Pookkillath, Alberto Roper Pol, Sourav Roy Chowdhury

In the last few decades, cosmology has been greatly enhanced due to a large number of precise observations, among which the EM ones have played a major role (refer to e.g. [605] for a recent review). Nevertheless, since the very first GW detection GW150914 [1], we got access to a new and uncharted observable that can help to shed light on the expansion history of the Universe by providing both information complementary to EM probes (e.g. [606–632]) and completely new potential observables [88, 620, 633–637]. Indeed, even if the current analyses of GWs mainly focused on the estimation of the present-day value of the Hubble parameter, $H_0$, eventually a more interesting set of observables will be the ones related to DE or modifications of GR at cosmological scales. At the background level, a non-trivial EoS for DE can give rise to deviations from the standard ΛCDM model, while in modified gravity theories perturbations can be different. In particular, tensor perturbations, i.e. GWs, can have a different propagation equation with respect to the one predicted by GR, in such a
way that both the velocity of GWs can be different from the speed of light (even though the observation of GW170817+GRB170817A [463] put very stringent constraints on modified gravity models), and the 'friction term' in the equation of motion for the tensor modes can vary, producing a difference between the luminosity distance estimated through GWs and the standard luminosity distance [638, 639]. Although with GW observations it might be difficult to improve on the accuracy of electromagnetic experiments for the estimation of the DE EoS (see, e.g. [640]), their complete independence from other measurement systematics makes GWs an extremely valuable new cosmological probe to solve tensions on cosmological parameters found by the current EM observations. Furthermore, the modified propagation can be accessed only through GW measurements, which are thus fundamental to constrain this kind of potential deviation from GR (that can result in effects even larger than the one associated to a non-trivial DE EoS).

In general, a GW signal emitted by a coalescing binary system, differently from EM cosmological probes, carries information on the luminosity distance to the source, which, combined with knowledge of its redshift, can be used to constrain the expansion history of the Universe through the $d_L - z$ relation, which in a flat $\Lambda$CDM model reads

$$d_L(z) = (1+z) \frac{c}{H_0} \int_0^z \frac{d\tilde{z}}{\sqrt{\Omega_{m,0}(1+\tilde{z})^3 + \Omega_{r,0}(1+\tilde{z})^4 + \rho_{\text{DE}}(\tilde{z})/\rho_0}}, \quad (3.10)$$

where $\Omega_{m,0}$ and $\Omega_{r,0}$ denote the present-day values of the density fractions for matter and radiation, respectively, while $\rho_{\text{DE}}(z)$ is the dark energy density as a function of redshift and $\rho_0 = 3H_0^2/(8\pi G)$. The crucial aspect in the GW context is then obtaining information on the sources' redshift. In the best-case scenario, the GW event is followed by a detectable EM counterpart which leads to a precise estimate of $z$, as we will discuss in the following. In case a counterpart is not available, GW detections can still be used for cosmological purposes through statistical methods. As first proposed in [606], it is possible to statistically obtain redshift measurements from potential host galaxies falling within the observed GW's localization error volumes exploiting galaxy catalogs. Also, a reconstruction of the intrinsic source-frame mass distribution of the observed mergers, in particular in the presence of sharp features, can lead to a statistical determination of the sources' redshifts, being the observed quantities in the detector-frame redshifted. Several recent studies have successfully tested these methods and their combination, as well as other similar ones, with both real data [616-621, 630, 636] and simulations [611, 613, 641-648]. For the former method to be informative, detections with narrow error volumes, both in the angular and distance components, are extremely valuable. Another independent approach to measure the expansion history of the Universe as well as the GW bias parameter is using the cross-correlation technique, which can measure the spatial clustering of GW sources with the galaxies [614, 623, 625, 629, 649-654]. This method can further explore weak gravitational lensing [624, 655, 656] and provide tests of GR modifications at cosmological scales in a model-independent way [651]. Other methods to employ GW detections for cosmological purposes include, e.g., exploiting multiple images of strongly lensed events [657-659]; having prior knowledge on the redshift distribution of the observed sources [660, 661]; using tidal distortions of NSs combined with EoS knowledge to extract information on the source-frame masses [610, 662, 663]; or measuring the angular barion acoustic oscillation scale from a high number of well localized events [664].
Already from the about 90 detections in the first three observing runs of the current GW interferometers, the potential of GWs for cosmology has been assessed through different techniques. From the observation of GW170817 and its EM counterpart, i.e. a 'bright siren' for which the distance is known from the GW signal and redshift from the host galaxy identification, it has been possible to obtain a few percent measurement of $H_0$ [665]. The estimate was $H_0 = 70_{-8}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$ with the dominant error contribution being the degeneracy in the GW signal between the source distance and the observing angle. Breaking the degeneracy using precise measurement on the observing angle obtained with very large baseline interferometry (VLBI) high angular resolution imaging of the radio counterpart of GW170817, the Hubble constant measurement improved to $H_0 = 70.3_{-5.0}^{+5.3} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [666] (see also [667] where using a different treatment for the peculiar velocity of the host galaxy it has been estimated $H_0 = 68.3_{-4.5}^{+4.6} \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $H_0 = 68.3_{-8}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$ with and without exploiting the VLBI information, respectively).

For GW-only events, the so-called ‘dark sirens’ without an EM counterpart and the host galaxy identification, statistical approaches can be adopted. Despite the low statistics, it has also been possible to get an interesting accuracy on the local expansion rate of the Universe [616–621], as well as other modified gravity observables [620, 634, 635, 637]. As an example, from the binary black-hole merger GW170814 in combination with the Dark Energy Survey it has been possible to set the constraint $H_0 = 75_{-32}^{+40} \text{ km s}^{-1} \text{ Mpc}^{-1}$ [617]. Using the sources in the GWTC-3 catalog, the LVK Collaboration obtained a constraint of $H_0 = 50_{-30}^{+37} \text{ km s}^{-1} \text{ Mpc}^{-1}$ adopting the statistical method relying on the source-frame mass distribution reconstruction and a POWER-LAW + PEAK profile, while a tighter constraint of $H_0 = 67_{-12}^{+13} \text{ km s}^{-1} \text{ Mpc}^{-1}$ has been obtained from the same sources with the statistical host identification technique [621].<fnref n="11" /> Combining with the posterior obtained for GW170817 and its associated counterpart, these constraints shrink to $H_0 = 68_{-8}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $H_0 = 68_{-6}^{+8} \text{ km s}^{-1} \text{ Mpc}^{-1}$, respectively. A more recent result obtained in [647] from the combination of these two techniques applied to the GWTC-3 events is $H_0 = 69_{-7}^{+12} \text{ km s}^{-1} \text{ Mpc}^{-1}$.

In the coming decades, we expect GWs to thrive in the context of cosmology, thanks to the increasing number of detections and a new generation of GW detectors, which on one hand aim to improve the reach and sensitivity of the current facilities, and on the other hand seek to open the door to new frequency bands. Being sensitive to GWs in the dHz band, a frequency region where both resolvable sources and stochastic backgrounds are expected to be observed, LGWA will be a valuable addition to the GW landscape.

In particular, in the bright siren case, we expect EM counterparts from CBCs in the Hz-kHz band to be produced by mergers involving at least one deformable object, such as a neutron star, thus coalescences at $\gtrsim \mathcal{O}(\text{kHz})$ frequencies, falling outside the LGWA band. However, LGWA is able to detect the early inspiral of such systems and can be crucial to localize the source in the sky with good accuracy, hence improving the chances of counterpart identification (see section 2.2.4). Even though the merit of joint GW+EM detections for cosmology is particularly high, systems with an associated counterpart are expected to be rare, with $\mathcal{O}(1)$ forecasted BNS multimessenger detections per year with LGWA (see section 2.2.1). It should be noted that, even though the rate of multimessenger

<footnote n="11">¹¹Notice that this value strongly depends on the assumptions for the BBH mass distribution.</footnote>
BNS events with LGWA is expected to be considerably lower than what will be achievable with 3G ground-based detectors, with tens to hundreds of expected multimessenger detections per year (see e.g. [67, 668]), the improved chances of counterpart identification thanks to the good LGWA localization can be valuable and will raise the number of bright siren events. Moreover, multiband observations of this kind of systems with LGWA and 3G detectors could be a powerful tool to study deviations from GR at the level of propagation of GWs through the Universe [88], with the possible counterpart identification adding even more information as compared to the BBH case.

In the dark siren case instead, at least for some of the coalescences falling in the band of terrestrial detectors, in particular at low redshifts, LGWA can play a crucial role, significantly shrinking the sky localization area thanks to the long time spent by the sources in the dHz detector band, and thus the huge effective baseline (see section 2.2.4). Combined with the accuracy in particular third generation terrestrial interferometers can have in the reconstruction of the luminosity distance, such detections can be a game changer for dark-siren GW cosmology, reducing the number of potential host galaxies in such small error volumes to only a few, and thus acting effectively almost as bright sirens. Moreover, the detections of MBH and IMBH mergers that occur in the LGWA band could be used for both the aforementioned methodologies, increasing the statistics and thus the constraining power, with the statistical uncertainty scaling as $\propto 1/\sqrt{N_{\text{events}}}$. Another interesting opportunity offered by these massive sources is the possible detection of EM radiation preceding or accompanying the gravitational signal from the environment surrounding the compact objects, making them bright siren candidates highly valuable for cosmology (see section 3.3.4). Finally, as shown in figure 7, LGWA will be able to detect coalescences of DWD and NSWD binaries up to $\sim 100$ Mpc. Thanks to the long time this kind of sources spend inspiralling in the detector's band, they can potentially be localized with very high accuracy, and provide a powerful tool to estimate the $H_0$ value in the local Universe, both as bright (see section 3.3.1 for a discussion regarding the EM transients these systems can give rise to) and dark sirens in combination with galaxy catalogs if counterpart detections are not available. The estimation of all the other cosmological parameters impacting the $d_L - z$ relation at higher redshifts coming from the other classes of sources discussed above would greatly benefit from this low redshift measurement.

Another observable and yet elusive source for current detectors is a SGWB of cosmological origin (CGWB). This refers to tensor modes possibly produced by various physical processes happening in the early universe, including, e.g., phase transitions, inflation, cosmic strings or primordial black holes (see e.g. [669–671] for reviews). It is, in a sense, the GW equivalent of the Cosmic Microwave Background EM radiation, whose discovery in 1965 has been one of the most relevant in cosmology. The detection of a CGWB would also bring outstanding scientific value, providing direct information about our Universe in its very first instants of evolution that would not be accessible by other means.<fnref n="12" /> The characteristic frequency $f_0$ of a CGWB at present time is determined by the Hubble rate at the time of generation of

<footnote n="12">¹²As studied for example in [672], an astrophysical GW background generated by stellar origin CBCs might dominate the SGWB also in the decihertz band. Sophisticated methods will have to be applied to reduce its contribution for CGWB observations [673–675].</footnote>
the background, redshifted to the present time [670]

$$f_0 = H_* \frac{a_*}{a_0} \approx 1.65 \times 10^{-7} \text{Hz} \frac{T_*}{\text{GeV}} \left( \frac{g_*}{100} \right)^{1/6}, \qquad (3.11)$$

where $T_*$ and $g_*$ correspond to the temperature scale and number of relativistic degrees of freedom at the time of generation, and $a_*$ and $a_0$ to the scale factors at the time of generation and today, respectively. In particular, if the source is causal and short, e.g., phase transitions, the resulting CGWB will peak at a frequency $f_p \sim f_0/(R_*H_*)$, where $R_*$ is the characteristic scale of the source, given as a fraction of the Hubble scale, $R_*H_* \le 1$. Hence, LGWA allows to explore e.g. phase transitions occurring in the early universe at temperature scales up to $T_* \sim \mathcal{O}(10^6)$ GeV.<fnref n="13" />

In figure 30 we show examples of possible CGWB generated by two different possible sources: cosmic strings and first-order phase transitions. Cosmic strings are line-like topological defects formed in grand unified theories by symmetry breaking following a phase transition [677–679] that can emit bursts of beamed gravitational radiation, whose incoherent superposition during the evolution of the Universe can give rise to a non-Gaussian SGWB [678] and can be used to probe particle physics beyond the Standard Model at energy scales not accessible by accelerators. We adopt the template developed in [680, 681] with a loop-size parameter $\alpha = 10^{-1}$ and different values for the string tension parameter $G\mu$ (related to the energy scale $\eta$ at which the strings are formed by $G\mu \sim 10^{-6}[\eta/(10^{16}\text{GeV})]^2$). In a first-order phase transition (FOPT), GWs can be produced by the collision of the broken-phase bubble walls [682–687], the production of sound waves [688–694], and the development of magnetohydrodynamical turbulence in the primordial plasma [695–703]. In figure 30, we show the CGWB produced by sound waves, which is expected to produce the dominant signal from a FOPT [681, 704, 705], based on the results of [693] for a FOPT strength $\alpha = 0.1$ and a bubble wall velocity $\xi_w = 0.4$.

With various models and sources that predict different trends and amplitudes as a function of frequency, the LGWA sensitivity in the dHz band could be extremely helpful to perform a CGWB detection, filling the gap between terrestrial detectors and LISA. Already as a single detector (in which case the form of the signal has to be known in order to extract it from the noise), LGWA could have an interesting sensitivity to SGWBs, since the minimum detectable GW background amplitude as a single instrument for a given SNR scales as $\propto S_n(f) f^3$ (with $S_n(f)$ being the detector PSD, [706]): comparing, for example, the sensitivity of a single LIGO instrument with the best forecasted O5 sensitivity at 300 Hz to LGWA-Si at 0.3 Hz we see that $[S_n(300 \text{ Hz})(300 \text{ Hz})^3]_{\text{LIGO-O5}}/[S_n(0.3 \text{ Hz})(0.3 \text{ Hz})^3]_{\text{LGWA-Si}} \approx 5 \times 10^4$.

Moreover, it could be possible to cross-correlate the output of LGWA with other non-dedicated experiments present on the Moon surface: their sensitivity could, of course, be lower than LGWA, yet the correlation of two or more detectors is much more promising for detectability, and no prior knowledge on the form of the signal is needed. In this context, it is also interesting to study the possibility of having a second antipodal detector consisting of a single seismometer with the same sensitivity of LGWA, which is a more optimistic scenario for

<footnote n="13">¹³Signals sourced by phase transitions at high temperatures $T_* \gtrsim \mathcal{O}(10)$ GeV may experience a different redshift, see, e.g., [676].</footnote>
<figure><img src="image_33.png" /><figcaption>**Figure 30.** Power-law integrated sensitivity curves [normalised to $h = H_0/(100 \text{ km s}^{-1}\text{Mpc}^{-1})$] for the LIGO-Virgo-KAGRA network with sensitivities representative of O5, ET+2CE network, LISA, and LGWA in two configurations considering two antipodal detectors. For LGWA we used sensitivities for a Nb detector plus an Si one, and two Si detectors, so to have a proxy of the output from correlation with a non-dedicated and a dedicated experiment, respectively. All the curves are obtained considering an observational time of $T_{\text{obs}} = 1$ yr and an SNR of 1. We further report examples of CGWBs generated by sound waves produced during a first-order phase transition (with different characteristic scales and temperatures, such that $T_*/(R_*H_*) = 10^6$ GeV, with a phase transition strength parameter $\alpha = 0.1$, and wall velocity $\xi_w = 0.4$) [693] and cosmic strings (with different string tension parameters $G\mu$) [680] observable at LGWA.</figcaption></figure>

SGWB searches. We show in figure 30 the power-law integrated sensitivity curves (introduced in [707]) computed as in appendix A of [67], for the LIGO-Virgo-KAGRA detector network with sensitivities representative of the O5 run; ET in the reference triangular configuration with 10 km-long arms in a network with two CE detectors, one with 40 km-long arms and one with 20 km-long arms, both located in the U.S.; LISA (with the procedure outlined in [708]); LGWA always considering the reference 4-seismometer configuration with Si sensitivity on one pole and another single detector on the other, both with a lower Nb sensitivity and with the same Si one to simulate a non-dedicated and dedicated experiment, respectively.<fnref n="14" /> From figure 30 we can appreciate how LGWA could, e.g., improve on the LVK results for cosmic strings [709] or access novel sources peaking in its frequency range.

Notice that, for antipodal detectors, in the frequency region covered by LGWA, the so-called overlap reduction function [710] of the detectors has a nearly constant value of 1. Indeed, using the results in [18, 36], it is straightforward to see that, for two antipodal seismometers, the overlap reduction function $\gamma(f)$ at frequency $f$ reduces to $\gamma(f) = 15 j_2(\Phi(f))/\Phi^2(f)$,

<footnote n="14">¹⁴Notice that, since in figure 30 we report a power-law integrated sensitivity curves for the full LIGO-Virgo-KAGRA network during O5, we do not observe the same ~$O(10^4)$ improvement reported in the text in the case of a single LGWA compared to a single LIGO.</footnote>
where $\Phi(f) = 4\pi f R/c$ (with R the Moon radius) and $j_2$ denotes the spherical Bessel function of the second order. Observe that $\Phi(f)$ is the phase difference between the two antipodal seismometers for a GW propagating along the line connecting them. In the frequency range relevant for LGWA (between 1 mHz and a few Hz) the function $\gamma(f)$ is, to a very good approximation, constant and equal to 1.

### 3.2.4 Fundamental physics with GWs

*Main contributors:* Parameswaran Ajith, Chandrachur Chakraborty, Bradley J. Kavanagh, N. V, Krishnendu, Prayush Kumar, Andrea Maselli, Suvodip Mukherjee, R. Prasad, Vaishak Prasad

**Parametrized tests of gravity.** Observations of low-frequency signals emitted by compact binaries evolving in the LGWA band represent golden sources for tests of gravity in the strong field regime and for searches of new fundamental physics [711, 712]. Among the variety of approaches developed to probe the existence of GR deviations, agnostic tests stand as a powerful and flexible tool, that has been extensively applied to actual observations [713, 714].

The ppE approach builds upon the parametrized post-Newtonian formalism [715–718], and introduces shifts in both the phase and the amplitude of the waveforms, which can be written in frequency domain as

$$h(f) = h_{\text{GR}}(f)(1 + \alpha u^{\beta})e^{i\delta u^{\gamma}}, \quad (3.12)$$

where $u \equiv (\pi Mf)^{2/3}$, $M$ is the binary chirp mass, $f$ is the Fourier frequency, and $(\alpha, \beta, \gamma, \delta)$ are the ppE parameters [719, 720]. $\beta$ and $\zeta$ identify the class of modification introduced, while $\alpha$ and $\delta$ control the magnitude of the deviation, and can be constrained by data. Bounds (or measurements) of such parameters can be eventually mapped to fundamental couplings of specific theories of gravity, effectively reducing the parameter space possible in alternative theories of gravity [721–723].

With agnostic tests being designed to modify the entire pN expansion of the waveform, LGWA would be particularly relevant to constrain the so-called pre-Newtonian (or *negative* pN corrections) effects, entering the signal before the quadrupolar contribution. A notable example is given by the −1PN dipolar emission, that arises in many extensions of GR which include extra fields coupled to the gravity sector [89, 724]. Such negative pN terms are more effective at lower frequencies, i.e. when systems orbit far from the coalescence [90, 92, 725]. Indeed, constraints on agnostic parameters would benefit from the thousands of cycles accumulated by stellar mass binaries in the deci-Hz band, allowing LGWA to provide smoking gun signatures on the existence of such new fields. LGWA would also significantly improve measurements of parameters that work at higher frequencies (*positive* pN corrections), that are more effective in the band of terrestrial detectors. In this case multi-band observations between LGWA and a 3G detectors like ET or CE would strongly reduce correlations among the waveform parameters, narrowing the bound on the possible GR shift.

The ppE formalism is flexible enough to incorporate a variety of effects deviating from the vanilla vacuum-GR scenario. Indeed, along with changes in the gravity sector, this
approach has been recently exploited to describe environmental effects, i.e. changes in the
waveform due to the presence of matter, with either a baryonic or a dark component, in
the binary surroundings [726–737]. Environmental signatures typically affect the signal at
very pre-Newtonian orders, and hence are best studied by low frequency instruments. In
this regard, deci-Hz detectors are expected to provide constraints on non-vacuum spacetimes
with exquisite precision [731].

**Testing the nature of black holes and other compact objects.** Gravitational wave signals emitted by merging binaries provide a new, golden tool, to probe the nature of BHs [738], exploiting observations of the inspiral and the post postmerger phases of the coalescence, and studying different but complementary signatures within the GW spectrum.

*Quasi normal mode oscillations.* In GR, the inspiral and subsequent merger of two BHs will result in the formation of a distorted black hole that will settle down to an equilibrium state given by the Kerr metric. The gravitational radiation from the final stages can be described by a superposition of the QNMs of the Kerr spectrum [739–742]. According to the no-hair theorem [743, 744] of GR, the oscillation frequencies and damping times of these QNMs can be characterized solely by the mass and spin of the remnant BH [745–747]. Thus, detecting multiple modes will provide crucial information about the nature of the remnant, and enable interesting tests of GR [748–760]. LGWA will be able to measure the QNMs of massive BHs ($M \sim 10^2 - 10^4 M_\odot$) with high SNRs, thus enabling precision BH spectroscopy. Deviations from the predicted structure can be mapped to specific beyond GR models or exotic compact objects [761–773]. Additionally, multi-band observations between LGWA and terrestrial detectors will allow us to simultaneously measure the masses and spins of the BHs in the binary as well as that of the remnant BH. This will allow us to do interesting consistency tests between these estimates [774–776].

*Spin-induced multipole moments.* Multipole moments provide yet another set of observables which can be exploited to test the nature of compact objects [777–781]. They affect the GW emission, and for BHs in GR their imprint on the signal is uniquely characterized by the BH mass and spin. However, other compact objects like neutron stars and boson stars, or BHs in alternative theories, will have spin-induced moments that depend on their additional properties (e.g., the equation of state of the object, or additional parameters in the theory). Measurements (or bounds) of these moments can be used to constrain the nature of compact objects or the theory of gravity [777]. The multipole structure of the compact object can also be affected by the presence of an external tidal gravitational field (e.g., produced by the binary companion). The corresponding deformations, quantified by the *tidal Love numbers* contain imprints of the internal structure of the compact object [738, 782, 783]. These effects are most accurately calculated for the inspiral part of the GW signal. By measuring the long inspiral signal from compact binaries, LGWA will be able to provide tight constraints on the nature of compact objects.

**Probing dark matter.** The presence of over-dense ‘spikes’ of particle DM around compact object binaries can lead to detectable effects on GW waveforms [726–729]. In particular, dynamical friction, the accretion of the DM particles, and the non-point-like gravitational
potential of the spike can alter the rate of inspiral and therefore the GW phase compared to the vacuum case [734, 784-788]. While the effect is typically small, these effects can accumulate over many orbits, leading to a roughly percent-level change in the number of GW cycles observed while the system is in band [734]. These dense DM spikes may arise from the adiabatic growth of astrophysical BHs at the centres of DM halos [789-791]. Alternatively, if PBHs are produced in the early Universe [792], they will naturally be surrounded by large overdensities of particle DM (assuming that the PBHs do not constitute all of the DM themselves) [793-795]. PBHs themselves might significantly contribute to dark matter [796], and they might have acted as seeds of today's supermassive black holes [797], whose early presence in the Universe — especially of those revealed by the latest JWST discoveries [798] — challenges models of their formation.

Feedback effects are expected to disrupt the DM spike if the secondary compact object is too massive [734, 799], which means that a sufficiently large mass ratio is required for the spike to survive around the primary BH. However, DM spikes around supermassive BHs are likely to have been disrupted by the motion of stars and gas in their environments [790, 800]. Intermediate-mass BHs instead are expected to live in more pristine environments [514] where a possible DM spike is more likely to survive until today. This points towards intermediate mass ratio inspirals as promising targets, with typical systems having $m_1 \in [10^3, 10^5] M_\odot$ and $m_2 \in [1, 10] M_\odot$. Such systems give rise to GWs in the range $f_{GW} \sim 0.1-1$ Hz in the last few years leading up to their merger, making deci-Hz observatories such as LGWA ideal probes.

The small DM-induced dephasing in IMRIs should be sufficient to detect the presence of a cold DM spike and to estimate its density profile [731, 735, 737]. In addition, the dephasing due to a spike of cold DM particle should be distinguishable from other possible sources of dephasing, such as that from an ultra-light boson cloud or from a baryonic accretion disk [801]. The detection of a dense DM spike would also shed light on the ultimate nature of DM, ruling out models which cannot achieve such a high density, such as light fermionic DM or self-annihilating DM [732]. Previous estimates of the detectability of this DM dephasing effect have generally focused on LISA, while we expect that for LGWA the enhanced sensitivity at decihertz frequencies should lead to even better prospects.

Observations of IMRIs by LGWA could also uncover sub-solar mass BHs. Such BHs could be of primordial origin (see, e.g., [416]) or could be produced by the accretion of particle dark matter into neutron stars or white dwarfs (dark matter transmutation; see, e.g., [802]).

## 3.3 Multi-messenger observations

The LGWA science case is particularly rich in astrophysical and matter phenomena. As a consequence, many of the GW sources are expected to emit detectable EM signals as well. Most of these events are specific to the deci-Hertz band. Some of these are low-probability events that may be extremely rewarding to detect. In this section, we outline the multi-messenger context of LGWA signals. The prospects of EM observations of binaries containing WDs is described in section 3.3.1. The detection of SNe together with associated GW signals would be a major breakthrough (section 3.3.2). Rare signals observed in EM and GW might come from TDEs of WDs near the horizon of BHs (section 3.3.3). Another multi-messenger study concerns MBH binaries in AGNs, where X-ray observations can complement the GW
detection (section 3.3.4). Finally, a new class of EM signals called QPEs are best explained by X-ray emissions from gas in E/IMRI systems (section 3.3.5). We refer to section 2.2.2 for an estimate of LGWA’s sky-localization capabilities, which plays a crucial role for most multi-messenger studies.

### 3.3.1 Compact binaries with white dwarfs and neutron stars

*Main contributors:* Michael Coughlin, Valeriya Korol, Javier Morán-Fraile, Tsvi Piran, Silvia Piranomonte, Jacopo Tissino

Compact binaries composed of NSs and WDs with short orbital periods are identified as prime candidates for LGWA (cf. section 3.2.2). An increasing number of these binaries are being detected at longer orbital periods just outside the sensitivity band of LGWA, millions of years before their eventual merger [803]. Concurrently, a rise in the number of transient events, which could be a result of the mergers of these same binaries, is being observed. Yet, the link between the progenitors and these transient events is often left ambiguous. LGWA offers a unique opportunity for multi-messenger observations both pre- and post-mergers of DWD and NSWD binaries. This section describes the electromagnetic observations of both NS/WD binaries and transients that may originate and outlines the synergies between LGWA and electromagnetic facilities.

**Electromagnetic observations of DWD binaries.** Recent advances in high-resolution interferometric observations over the past decade have demonstrated the ubiquity of multiple star systems [804]. Given that WDs represent the natural end stage of low-mass star evolution (~95% of all stars), double white dwarf binary systems (DWDs) should be abundant in nature. Indeed, the Galactic population of DWDs is projected to number in the hundreds of millions [805]. However, EM observations of these systems is challenging due to the inherent physical characteristics of WD stars, which are characterised by a mass comparable to that of the Sun but compressed into the size of an Earth-like planet [486]. Such a small size makes WDs inherently dim (with absolute magnitude in the range 10–14 mag in the Sloan g-band). While the pronounced gravitational force of these extremely dense stars leads to significant pressure broadening of the observable (spectroscopic) lines, gravitational settling also results in predominantly light elements (mainly Hydrogen and Helium) being most easily observed in their atmospheres.

The most effective detection strategy is to analyse the compact binary orbital motion via spectroscopy. By identifying radial velocity variations of the order of a few ×100 km s⁻¹, it is possible to identify a short period DWD binary, which photometrically resembles a single WD. Over the past two decades, advances in large spectroscopic surveys have enhanced the observation of DWD systems. The Supernova Ia Progenitor survey [806] targeted around 1000 bright WDs, confirming several dozen DWDs [475]. More recently, [807], aiming at the extremely low-mass WD (< 0.3 M☉), identified an additional hundred systems. However, spectroscopic surveys like these are relatively limited both in brightness and in numbers, motivating the use of time-domain photometric surveys to discover DWDs.

Time-domain large-scale photometric surveys search for these systems by observing eclipses or light curve modulations. Though less efficient, they may also help to evaluate
the efficiency of more common spectroscopic identification techniques, by comparison of estimated rates and systematics. Surveys such as the Asteroid Terrestrial-impact Last Alert System (ATLAS) [808] and the Zwicky Transient Facility (ZTF) [809], among others, are beginning to detect WD binaries regularly, with spectroscopy following for confirmation and characterization [810]. In addition, DWD binaries can be found in exoplanet-focused time-domain photometric surveys, as exemplified by detections from *Kepler* and *TESS* space missions [811–813]. In the near future, the *Gaia* survey [814] is poised to offer new insights into the field [815]. Additionally, state-of-the-art facilities like the Vera C. Rubin Observatory [816], along with tools specifically tailored for spectroscopic follow-ups like SOXS [817], promise to significantly contribute to the area. They are anticipated to yield hundreds, if not thousands, of new identifications [472, 494]. This wealth of data will facilitate the selection of a subset of short-period, accurately measured systems for dedicated monitoring with GW observatories.

Moving to higher energies, DWD binaries are difficult to detect since they can sustain only moderate accretion, if any, to feed high energy emission. Still, UV and soft X-rays observations for those relatively nearby systems detectable with present-day instruments are useful, for instance, to better constrain the spectral energy distribution of the components of the binary system. In turn, this might allow to better determine the size of the sources and characterize the evolution of the binary. The best example is likely RX J0806.3+1527 [818], modeled as a double degenerate binary with orbital period of 321 s, first identified analyzing ROSAT data [819]. In this system, mass transfer should come from a Roche lobe filling WD to another more massive WD. This kind of systems are also known as part of the AM CVn category, and are of particular interest for GW astronomy since the condition that a companion star fills its Roche lobe translates into a short orbital period, i.e. less than ~2 hours. Several tens of these binaries are presently known in this category [803, 820]. GW astronomy could also provide a reliable way to measure mass transfer and other system parameters to better constrain their still poorly known formation channel(s) [821].

Independently of the identification scenario, currently, only a couple hundred DWD systems are known. Given the variability in detection techniques, ranging from instrument resolution to sample definition, it remains challenging to derive statistically significant conclusions about the Galactic DWD population. However, recent estimates indicate that the DWD fraction stands at roughly ~6%, aligning with predictions from population synthesis [822]. Out of these, merely around ~10% [823] exhibit sub-hour orbital periods. These particular systems are bound to merge within a Hubble time due to the emission of GWs.

**Electromagnetic observations of NSWD binaries.** Being even more challenging to observe than DWD systems, NSWD binaries are also less numerous [278, 824]. However, there are a few ways that NSWDs can reveal themselves. For example, when a NS accretes from a WD companion, these systems are visible as Ultra-Compact X-ray Binaries (UCXBs). During the accretion process, the infalling material emits high-energy X-rays, making them visible in X-ray wavelengths. Additionally, if the NS behaves as a radio pulsar, its radio emission can be picked up as periodic signals, revealing its presence. However, Doppler smearing of radio pulsations occurs when the combined effect of the binary system's motion and the pulsar's inherent spin blurs the received radio signals. This smearing can introduce a bias against the detection of rapidly spinning millisecond radio pulsars in acceleration
searches. Recently, Pol et al. [825] have yielded accurate models of the observed binary pulsar population, highlighting a high likelihood of detecting NSWD systems with orbital periods under 15 minutes with Arecibo-like radio telescope surveys. This probability is anticipated to increase with the forthcoming Square Kilometre Array [826]. Moreover, they argued that unequal mass NSWD systems appear more detectable than those with near-equal masses. However, these binaries start interaction at orbital periods between 25–15 minutes, which can impede radio detection as discussed above.

**Transients that may arise from DWD/NSWD binaries.** Mergers involving double WD binaries and NSWD binaries can lead to explosive or transient events and the formation of exotic objects that typically do not emerge from the evolutionary path of single stars. Many studies have delved into the mergers of double WD binaries, mostly in the context of potential SN-Ia progenitors, as reviewed by various authors such as [278, 827–830]. These binaries are created through common envelope evolution in binary systems, and in the closest double WD binaries, the emission of GWs causes their orbits to shrink and initiates mass transfer between the WDs within a Hubble timescale. The critical factor influencing the final result is whether mass transfer remains stable or diverges, potentially leading to the tidal disruption of the donor star and binary merger [831]. Mass transfer's stability depends on the response of the WDs to the process, their masses, and various angular momentum transport mechanisms, such as torques due to accretion discs or tidal bulges [832, 833]. Subsequent evolution after WD mergers can yield diverse outcomes, including He-rich hot subdwarfs, R Coronae Borealis stars, massive carbon/oxygen or oxygen/neon WDs, SN-Ia, accretion-induced collapse to a neutron star or even a black hole. A chart summarizing these potential outcomes of WD mergers can be found in figure 3 of [834]. The case of SN Ia holds broader historical significance and particular scientific interest for LGWA, and is therefore discussed in detail in section 3.3.2.

Mergers involving NSWD and BHWD binaries are also expected to fall within the LGWA sensitivity band and generate electromagnetic transients. However, simulating theoretical predictions about their observational signatures to estimate them numerically presents more challenges. The main challenge lies in dealing with a wide range of scales and complex physical processes inherent in this problem. This is in contrast to mergers involving objects of similar size, such as DWD or NSNS/BH, which are more manageable and have more well-developed observational predictions, as demonstrated in studies like [508, 835, 836]. The merger events involving NS or BH with a WD have been extensively studied through various hydrodynamical and nuclear-hydrodynamical simulations. Some notable references in this research include [274, 275, 280, 282, 837]. It is generally anticipated that all NSWD mergers result in unstable mass transfer [277]. In such case, the WD is tidally disrupted on dynamical timescales, forming an extended debris disk around the NS. The evolution of this disk is primarily governed by viscosity, but nuclear burning also plays a significant role. Together, disk viscosity and nuclear burning drive outflows throughout the disk [274]. Nuclear burning within the disk progresses steadily, potentially with a weak detonation, and it results in the production of small amounts of $^{56}$Ni, typically at most $10^{-2}$ M$_{\odot}$ [837]. This limited amount of $^{56}$Ni is expected to lead to optical transients, but they are significantly fainter compared to typical type Ia supernovae, which generate over an order of magnitude more $^{56}$Ni. The
transients resulting from NSWD mergers could potentially represent an entirely distinct class of supernovae. These events might be observable primarily in nearby galaxies using large telescopes or, perhaps, with next-generation survey instruments like Rubin-LSST.

**EM-LGWA synergies.** Binaries with a WD component are not loud emitters of gravitational waves, due to their low masses and long orbital period. Most such binaries will be detected in their inspiral phase integrating the quasi-monotonic signal over years, similarly to what is done for continuous wave searches by current GW detectors. A few of these detected binaries will merge during the LGWA mission lifetime. The presence of eccentricity — expected for NSWD binaries if the NS forms after the WD [278] — gives rise to higher-frequency harmonics of the emission [706, 838], which boost overall emitted power as well as placing it closer to the most sensitive band for LGWA. As a summary of what is discussed in section 2.2.1, deci-Hz GW sources in our own Milky Way are very likely to be observed regardless of their orientation, while sources outside our galaxy might be seen mostly if they have favorable orientations and masses.

Multi-messenger detections of WD mergers within our galaxy should not be expected, even though the Milky Way hosts an estimated $O(10^8)$ DWD and $O(10^7)$ NSWD binaries [484]. This is because only a small fraction exists as short-period binaries. Hence, LGWA's capability to see binaries with WDs out to a few 10 Mpc (see section 2.2.1) is crucial for the success of a EM-GW multi-messenger campaign. For reference, the detection rates of DWD merger derived from SN Ia observations are estimated in section 3.2.2. More details of a DWD and SN Ia association and a possible coincident detection can be found in section 3.3.2.

Nevertheless, pinpointing even a limited number of ultra-short period DWD/NSWD binaries can yield significant astrophysical insight. Optical follow-ups could be essential in resolving parameter ambiguities that arise from relying solely on GW signals. These parameters include system mass, distance, and orbital period, especially if light-curve modulations are discerned. Collectively, such observations can elucidate various domains, from white dwarf structures and binary stellar evolution to accretion physics, general relativity, and the overarching structure of our galaxy. The association of a gravitational event with an electromagnetic counterpart, such as SN-Ia explosion, would be a groundbreaking discovery, much like for the first multi-messenger observation of binary NS merger GW170817 [839].

Concerning NSWD mergers, in the event the NS does not collapse into a BH post-merger, Morán-Fraile et al. [282] estimate jet velocities similar to those seen in galactic microquasars ($\gtrsim 0.9c$), leading to an event spectrum that initiates with a soft X-ray or UV flash and transitions into lower frequencies, resembling fast blue optical transients (FBOTs, [840]), potentially associated with NSWD mergers. The merger could generate a prolonged, bright radio signal, with the transient's luminosity potentially exceeding typical kilonovae due to the ejecta's mass and velocity, rather than r-process yields, which are obscured in kilonovae by lanthanides. If the NS transitions into a BH, this could result in ultrarelativistic jet speeds, leading to long or ultra-long GRB-like transients, marking a significant observational shift
### 3.3.2 Supernovae

*Main contributors:* Ferdinando Patat, Stefano Benetti, Marica Branchesi, Enrico Cappellaro, Kiranjyot Gill, Elisabeth-Adelheid Keppler, Francesco Longo, Sourav Roy Chowdhury, David Vartanyan

**Thermonuclear Supernovae.** Thermonuclear explosions, better known as Type Ia Supernovae (SN Ia), play a crucial role in modern cosmology. In addition to providing the first evidence that the expansion of the Universe is accelerated by some sort of dark energy, they recently led to a tension with the value of the Hubble constant derived from the Cosmic Microwave Background (CMB) modelling (see [841] for a review). Despite their importance, the nature of their progenitors is still unclear and two alternative scenarios have been proposed: the single-degenerate (SD) scenario and the double-degenerate (DD) scenario (see section 3.2.2). The pieces of evidence collected so far in favour of one or the other option are all indirect. As examples of relevant results, we can list: i) the presence [842] or absence [843] of signatures of circumstellar gas surrounding the progenitor that favour SD and DD, respectively; ii) the failed search for a surviving companion of historical SN Ia in our Galaxy, favoring DD [844]; iii) the failed search of a precursor star in pre-explosion archival images [845] favoring DD; iv) the search for candidate DD systems that did not find a sufficient number of short period, massive WDs to account for the observe SN Ia rates [475] favoring SD, or otherwise sub-Chandrasekhar explosions; v) the distribution of delay time from star formation to explosion derived from statistics of SN Ia rate that tends to favor the DD scenario [497], although the strength of this argument is debatable [846].

The bottom line is that, after decades of research, none of the results is sufficiently conclusive and, in some cases, the findings appear to be even in contradiction. Moreover, observations have shown that about 30% of the discovered SNe Ia largely deviate from the properties of “normal” thermonuclear events, such as peak luminosity, light curve morphology and spectral features (for a recent review see [847]). The rising era of GW astronomy opens a new avenue for obtaining independent and direct insights to the SN Ia progenitors’ nature, which remains one of the burning, open questions in modern astrophysics [848].

Arguably, the ultimate proof of the DD-SNIa connection would be the observation of the merging event followed by the detection of its electromagnetic (EM) counterpart. However, while in some theoretical scenarios the SN explosion and hence the appearance of the EM counterpart, follows a few seconds after the merger [508], others allow for a long delay between the merger and the explosion, extending to $10^4$ yr [849]. This implies that, even when a DWD merger with the appropriate mass occurs and its GW emission is detected, it may not be associated with a prompt SN Ia EM event. Hence, missing the long-sought and much-desired smoking gun signature. The comparison between the known rate of SNe Ia and the rate of GW sources exhibiting properties expected for merging DD systems does not require the DD-SN Ia association to answer this key question (as described in section 3.2.2). As shown in section 2.2.1, LGWA can observe DWD signals out to several tens of Mpc. Using SN Ia observations to estimate the corresponding DWD merger rate (assuming that all SN Ia are produced by DWDs), we found in section 3.2.2 that about 20 DWDs producing SN Ia would
be observed in 10 years with LGWA. This would be enough detections to make a meaningful comparison of rates and infer a possible DD progenitor scenario.

Interestingly enough, the simultaneous detection of a “peculiar” SN Ia in both GW and EM domains is expected for thermonuclear events arising from WD-WD systems originating from a CO-WD and a He-WD. These objects are anticipated to explode via the detonation of a thin He-layer piled-up on the CO-WD through a stable mass transfer that is driven by GW emission with maximum expected frequency of several mHz (see [850] for details). Multi-messenger astronomy could help determining the contribution of double-detonating WDs to the observed SNe Ia rate and, more broadly, allow one to infer the chirp masses for some of the transient, hence constraining the SN progenitor.

**Core-Collapse Supernovae.** Following the gravitational collapse of a massive star ($M \ge 8M_{\odot}$), the inner core rebounds and collides with the infalling stellar material producing a shock wave (see [851] for a general review). The shock wave tends to stall within the infalling material. Extensive theoretical work done in recent years has demonstrated that it is the neutrino heating coming from the accreting proto-NS (coupled to the neutrino-driven turbulent convection between the shock cavity and the stellar surface) that re-powers the shock and leads to subsequent explosion ([332, 341, 852] and references therein). The shock break-out, i.e., the moment when the blast wave reaches the stellar surface, occurs minutes to days post collapse, and shortly after starts emitting electromagnetic radiation, appearing as a SN. The star is destroyed leaving behind a compact remnant.

It is clear that CCSNe are natural-born candidates for multi-messenger detection, as they are sources of EM, GW radiation and neutrinos. The CC SN1987A indeed marked the birth of neutrino astronomy [853, 854]. The temporal coincidence of the EM and neutrino signal produced by SN1987A has clearly shown the association between the neutrino emission and the (supposed) core-collapse, hence providing direct support to the underlying theory.

Although more frequent than SNe Ia, CCSNe are rare events. Their estimated rate for the Milky Way is $1.63 \pm 0.46 (100 \text{yr})^{-1}$, which corresponds to an average recurrence time of $61_{-14}^{+24} \text{yr}$ [855]. As described in section 3.2.1, key for the detectability of the GW signal from CCSNe is the memory effect, which produces relatively large amplitudes below 10 Hz and into the decihertz band. In this way, the LGWA coupled with other future, planned GW detectors will open the possibility of observing these garden-variety events using all three messengers: EM, GW, neutrinos. Clearly, the simultaneous detection of all three would be the next watershed detection in the field of GWs.

### 3.3.3 Tidal disruption events

*Main contributors:* Martina Toscani, Francesca Onori, Elisa Bortolas, Marica Branchesi, Roberto Della Ceca, Giovanni Miniutti

Tidal disruption events (see section 3.2.1) are especially interesting transient phenomena for several reasons. First of all, they are exquisite EM sources, as they produce very bright flares (luminosities of order of $10^{43-44} \text{ erg s}^{-1}$) in different bands of the EM spectrum, extending from X-rays to radio wavelengths (for recent reviews see [856–858] and references therein). Thanks to this powerful emission, they have been an unparalleled tool to unveil the
presence of otherwise quiescent MBHs in the cores of galaxies. In addition to this, they also emit astrophysical neutrinos during the later stages, when they could trigger the formation of jets (for details see [859]). Finally, they are expected to emit GWs in the low frequency regime $f_{gw} \sim 10^{-4} - 10^{-1}$ Hz [297].

While the majority of EM observations of these events suggest that TDEs typically occur with MBH in the mass range of $\sim 10^6 - 10^8 M_{\odot}$, a particularly fascinating prospect lies in the detection of a TDE involving a WD around a MBH in the intermediate-mass range, $\sim 10^3 - 10^5 M_{\odot}$. These occurrences have proven challenging to detect thus far, despite some suggestive clues [860–862]. Observing such a WD-TDE would be a smoking gun of the existence of these elusive IMBHs [295], which have remained difficult to detect until now. It is worth noting that these WD-TDEs represent the best TDE sources of GWs in the LGWA frequency band.

Another interesting scenario involving tidal disruptions is the case of repeating partial TDEs, where a star, likely in an evolved state, follows a highly eccentric bound orbit and loses portions of its envelope during each pericenter passage due to the gravitational forces of the central MBH. This results in relatively short-lived, quasi-periodic electromagnetic flares, driven by accretion and/or shocks. Partial TDEs are considered to be the prevailing explanation for a new class of detected EM nuclear transients, the so-called repeating nuclear transients (RNTs), a small number of which has been observed in the optical (e.g. ASASSN-14ko [863, 864]) or X-ray bands (e.g. eRASSt J045650.3-203750 [865] and Swift J023017.0+283603 [866, 867]).

Similarly to the case of X-ray quasi-periodic eruptions discussed in section 3.3.5, RNTs are likely to represent the EM counterpart of extreme mass-ratio binary systems, opening up a possible window for multi-messenger synergies. If RNTs are indeed due to periodic mass transfer events at pericenter from the envelope of evolved donor stars, they are likely to evolve into MBH-WD (or WD-like) extreme mass-ratio binary systems, once the whole envelope has been stripped off.

**Electromagnetic emission from TDEs.** TDEs were anticipated to exhibit exceptional brightness in X-rays [285, 288]. Indeed, the first candidates [868, 869] were revealed as luminous X-ray sources in the ROSAT all sky survey archive and subsequently other X-ray TDEs were identified thanks to dedicated searches or serendipitous discoveries by using CHANDRA, XMM-Newton and Swift satellites [870–874]. However, over the past decade, wide-field optical transient surveys such as ASASSN, ATLAS, Pan-STARRS, iPTF, and ZTF have played a crucial role in making the optical band the primary channel for discovering TDEs. Consequently, the sample of TDEs has rapidly grown from few candidates to tens of confirmed TDEs, revealing a heterogeneous population of transients and enlightening a number of well established key observational properties both in optical and in X-ray bands [857, 858].

Flares produced by WD-TDEs are instead relatively rare with respect to that emitted by TDEs involving main sequence stars (MS-TDEs) and, although it has been currently reached a TDE discovery rate of $\sim 10$ events per year, no confirmed WD-TDEs have been reported so far, and only a few possible candidates. However, recent theoretical studies and simulations have been able to accurately predict their expected EM observational signatures [875]. In the following, we briefly review the main properties of their EM emission distinguishing between MS-TDEs and WD-TDEs.
MS-TDEs. MS-TDEs manifest themselves as large amplitude blue flares (g-r<0) in the core of quiescent galaxies (with an overabundance of TDE detection in post-starburst/E+A host galaxies), characterized by a roughly constant optical colours. The light-curves usually rise to the peak in approximately <30 days and show a power-law decline broadly consistent with the ~t⁻⁵/³ law [876] on time-scales of months to years. They are characterized by hot and near-constant black body temperatures (TBB ~10⁴ K) with a lack of cooling in the post peak phase [877]. X-rays detection of optically selected TDEs are rare, with only few exceptions, also including some events showing delayed X-ray flares (e.g. ASASSN-14li, ASASSN-15oi, AT2019dsg, AT2019qiz, AT2019azh and AT2017gge [878–883], respectively). Such an observational dicotomy has been explained as the result of the intrinsic physical properties of the EM emitting region coupled with viewing angle effects (see the TDE unification model proposed in [884]). When detected, the X-ray emission of MS-TDEs is typically in the soft energy band (0.1-2.5 keV), with peak luminosities between 10⁴² and 10⁴⁴ erg s⁻¹ and extremely soft spectra near the peak, well modeled by power-law with spectral indices Γx=3-5 (or, alternatively by blackbody model with temperatures kTbb=0.04-0.12 keV). Some events have shown a spectral hardening with time. The overall lightcurve decline follows quite well the ~t⁻⁵/³ power-law, but some events have shown also a fast variability on timescales of minutes to hours (see the reviews of [885, 886]). Notably exceptions to this picture are the so-called relativistic TDEs (AT2022cmc [887], Sw J1644+57 [888], Sw J2058.4+0516 [889] and Sw J1112.2-8238 [890]), extremely luminous, hard X-ray events (Lx ~10⁴⁷⁻⁴⁸ erg s⁻¹ at peak) in which the detection of a non-thermal component has been explained with the launch of face-on relativistic jets.

Optical spectroscopy is required to unambiguously identify a TDE. The early spectra
are dominated by a strong blue continuum with superimposed very broad (FWHM~10⁴
km/s) H and/or HeII λ4686 emission lines, with different strengths and relative ratios [891].
The line profiles are of pure emission with the FHWM decreasing with time. However, line
asymmetries, double peaked line profiles and outflows components have been reported for
some events. Recently, the notable discovery of TDEs exhibiting broad Bowen fluorescence
emission lines such as O III 3760 and N III 4100,4640 (iPTF15af [892], iPTF16fnl [893],
AT2018dyb [894]) have led to the mapping of the TDE spectral diversity into four main
sub-classes, the He-rich, H+He-rich, the H-rich and N-rich TDEs [857], and represented a
strong evidence for the presence of an obscured and reprocessed EUV/X-ray emission. In
figure 31 a compilation of continuum subtracted TDE optical spectra is shown.

WD-TDEs. The WD-TDEs EM emission differs from that observed in MS-TDEs since the WD disruption is expected to induce a super-Eddington accretion phase, which in turn results in the lunch of relativistic jets. Moreover, the MBH gravitational influence on a compact object such as a WD may produce an additional source of energy in case of deep encounters (β >3, with β defined as the ratio of the tidal radius above the stellar pericenter). In particular, the strong tidal compression experienced by the WD at the first pericenter passage, could results in a thermonuclear runaway reaction before the disruption and can trigger an optical transient powered by the radioactive decay of ⁵⁶Ni. The results is a highly asymmetric SNIa-like event [875]. Given the involvement of IMBHs in this process, also the host environment is
<figure><img src="image_34.png" /></figure>

**Figure 31.** Sequence of continuum subtracted TDE optical spectra. Very broad H and/or HeII emission lines are observed with different relative ratios. The TDE sub-classes are highlighted by different colors: TDE He (in blue), TDE H-He (in purple), TDE H (in red). Those showing additional NIII features are referred N-rich/Bowen TDEs. Reproduced from [857], with permission from Springer Nature.

different with respect the MS-TDEs, with WD-TDE expected to be detected either in the nuclei of dwarf galaxies, or in the outskirt regions of galaxies, or even in globular cluster.

The observational properties of this class of transient are highly viewing angle dependents. For systems observed face-on, a luminous X-ray jet precedes an optical afterglow, which is followed by the SNIa-like transient. After the thermonuclear transient starts to fade, a radio afterglow arise. In case of off-axis observers, instead, the early time emission is dominated by a soft X-ray component coming from a newly formed accretion disk, the SNIa-like transient outshines the optical afterglow and it is in turn followed by the radio afterglow [895]. When the high energy emission is detected, it is extremely luminous (with typical peak jet luminosities of ~$10^{47-50}$ erg s⁻¹), fast rising (with timescales of ~$10^{2-4}$s) and long lasting (with a super-Eddington accretion phase lasting months to years) [895].

As introduced before, in the case of WD-TDEs there are two sources behind the UV/optical emission: the thermonuclear reaction and the ionization of the stellar debris.

The transient resulting from the thermonuclear runaway reaction is expected to mimic the optical emission of SNIa but with lower luminosities ($M_B$ between -16 and -18 mag,
<figure><img src="image_35.png" /></figure>

**Figure 32.** EM observational features expected in the case of a WD-TDE involving a $M_{WD}=0.6$ WD and a $M_{\odot}$ and a $M_{\bullet} = 10^3 M_{\odot}$ IMBH. Times are from the WD periaurse passage. Dot-dashed lines indicate the signal seen by an observer placed along the jet axis, while solid lines refer to an off-axis observer. Reproduced from [875], with permission from Springer Nature.

$L_{bol} \sim 10^{40-42}$ at peak), bluer colors (g-r<0) and a faster time evolution (rise to the peak in ~5-10 days). In particular, the early UV/optical spectra are expected to be mainly featureless and dominated by a hot blue continuum, with the characteristic spectral features developing after ~15 days from the light-curve peak [875]. At this phase the spectra broadly resemble those of SNe, with broad absorption features and P-cygni profiles in correspondence of Si II, Ca II and Ti II (the lines intensities and widths have a viewing angle dependence). Key distinctive features are the presence of Doppler shift in the spectral lines time series (up to ±12,000 km/s), which is the result of the WD-TDE unbound ejecta orbital motion, and the detection of an X-ray emission [895].

The ionization of the stellar debris tail will produce broad emission lines such as C IV $\lambda$1550; C III$\lambda$977; [OIII]$\lambda$4959,5007 and [O III]$\lambda$4636, but are expected to appear in the UV/optical spectra at very late times (>100 days), when the WD-TDE is in the sub-Eddington accretion phase [276]. Their strength should decline with time following the $\sim t^{-5/3}$ law, with the [OIII]$\lambda$5007 having the slowest decline and thus to be still visible in the optical spectra taken a decade after the WD-TDE occurrence.

Although no confirmed WD-TDE has been reported so far, some candidates have been proposed such as the class of relativistic TDEs [860], fast X-rays transients [896, 897] and Ca-rich transients [898]. However, their nature is still highly debated.
Additional ways to identify candidate TDEs are the detection of reverberation signals caused by the interaction of the TDE EM prompt emission with a host environment enriched in dust and gas (see the review of [899]). In particular, the *WISE* [900] and the more recent *NEOWISE* [901] projects turn out to have a crucial role in finding the mid-infrared echoes resulting from the re-radiation of the TDE emission by dust [902]. Furthermore, the remarkable detection of transient high ionization coronal emission lines in the late-time optical and IR spectra of the TDE AT2017gge [883] represented the first observational evidence of the connection between the occurrence of a TDE and the presence of these emission lines in inactive galaxies. This finding strongly supporting the theory that extreme coronal line emitter (ECLE) galaxies may have indeed hosted a TDE in the past [903, 904].

**Neutrino emission from TDEs.** The idea of TDEs as neutrinos sources was first proposed after the discovery of the peculiar class of jetted-TDEs [905–907]. In particular, the neutrinos production was considered in the framework of the jet models as the result of different state in the accretion disk evolving from the debris circularization of stellar debris to super- and sub-Eddington to radiatively inefficient accretion flows and/or of the interaction of the TDE ejecta with the hosting circumnuclear environment [908, 909].

Indeed, ZTF EM follow-up searches of IceCube alerts identified TDE AT2019dsg [910] as a possible optical counterpart of a very high energy neutrino (~0.2 PeV, IceCube event IC191001A), which represents the first multi-messenger observation of a TDE. Later, other two TDE candidates have been associated with two high-energy neutrino events: AT2019fdr (IceCube event IC200530A [911]) and AT2019aalc (IceCube event IC191119A [912]). Interestingly, all the three neutrino-emitting TDEs discovered so far show some common observational features, such as the delayed neutrino signal (~100 days after the BB peak), a delayed strong dust echoes in the IR and the X-ray detection. This can be ascribed to the physical properties of the system after the star disruption and raise important question on the mechanisms behind the neutrino production itself [913]. However, it should be noted that recent reanalyses of IceCube data suggest that the coincidences between TDE and neutrino observations are consistent with background [914].

**Multimessenger perspectives and observational strategy.** GW-detected TDEs will mark the moment of disruption of the star, otherwise undetectable. Such a detection could serve as crucial information for telescopes to know where to search for subsequent EM/neutrino counterparts, which may arise within minutes to days after the disruption phase (depending on the exact onset of the accretion process).

Multimessenger detections of TDEs open the possibility to perform exciting scientific analyses. For example, measuring the time delay between the GW signal and subsequent EM flares would allow us to discriminate among different type of TDE EM mechanisms, to date still debated [915]. Moreover, the combined observation of these signals will allow us to constrain cosmological parameters, e.g., the Hubble constant [916, 917].

Given these exciting perspectives, it is important to outline a strategy for the multimessenger detections of these sources. First, the GW identification from a TDE can allow us to localise an area in the sky where to look for the expected EM counterpart. In particular, in the case of MS-TDEs, their identification as EM counterparts could come within weeks-months
from the GW alert, considering the delay between the disruption of the star (where the main GW burst is produced) and the beginning of the accretion. In this case, there is plenty of time to start promptly a multi-wavelength follow-up not only to identify the transient as a TDE candidate, but also to characterize it (i.e., photometric analysis to derive BB properties, a spectral sequence to monitor the development of broad lines and their profile and a proper X-ray follow-up). In particular, for MS-TDEs we have observational evidence of their overabundance in post-starburst/E+A galaxies. Thus, blue nuclear flares from such galaxies placed in the GW localization area would be the first objects to look for.

Given the short time delay between GW production and EM flares in the case of WD-TDEs, EM emission could actually be used as an alert for GW counterparts. Indeed, detected electromagnetic emission could be used as valuable information to analyse the interferometer's data stream and to look for GW signals compatible with the properties of the observed system.

Moreover, we recall that GW emission form TDEs is also possible during the circularization phase, that is as the same time as the production of EM radiation. Yet this GW emission is roughly 2 orders of magnitude weaker than the one emitted during the disruption phase [297].

It is worth noting that some late-time identifications of TDEs as EM counterpart of GW signals may come from the detection of mid-infrared echoes and/or transient high-ionization coronal emission lines in quiescent galaxies inside the GW localization area, making late-time searches important.

### 3.3.4 Intermediate and massive black hole binaries

*Main contributors:* Paola Severgnini, Cristian Vignali, Elisa Bortolas, Valentina Braito, Roberto Della Ceca, Alessia Franchini, Giovanni Miniutti, Deeshani Mitra, Alessandra De Rosa, Roberto Serafinelli, Jacopo Tissino

**Astrophysical relevance.** In the current multi-messenger era, the search for and the characterization of intermediate-mass (IMBH, $100 M_{\odot} < M_{BH} < 10^5 M_{\odot}$) and massive black hole (MBH, $M_{BH} \ge 10^5 M_{\odot}$) binaries are among the hottest and most challenging topics of modern observational astrophysics. Theory predicts that super-Eddington accretion onto stellar BHs (e.g., [918]) and a combination of stellar collisions, star-BH interactions, and BH-BH mergers (e.g., [404, 919-923]) are the main channels for the formation of IMBHs, that will then become the seeds for MBHs through further accretion and mergers (see section 3.2.2 and [516, 924, 925]). Since most galaxies harbor a central MBH, the current $\Lambda$CDM cosmological paradigm predicts a large population of MBH binaries as a natural outcome of galaxy mergers. These systems, which are amongst the loudest emitters of GWs, will eventually coalesce by forming a more massive BH. Thus, detecting and tracing the evolution of IMBH and MBH binary systems across cosmic time would be the key to understanding the hierarchical structure formation, the growth and demographics of BHs, and the accretion and feedback that regulate the MBH/galaxy interplay.

Despite their relevance, both IMBH and MBH binaries remain observationally very elusive. Shortly, upcoming facilities at various electromagnetic bands are expected to play a pivotal role in building up catalogues of EM-detected MBH/IMBH candidates for GW follow-up (see Electromagnetic emission). GWs will open the way to unambiguously recognize
<figure><img src="image_36.png" /></figure>

**Figure 33.** *Residence time*, i.e. the characteristic time a binary spends in each logarithmic bin of separation, as a function of the separation itself; different lines refer to different MBH masses. The residence time is computed as the separation *r* of the binary divided by its time derivative, *r*/(dr/dt). In the transition between stellar hardening and GWs, the presence of a circumbinary disc can shorten or perhaps enhance the residence time; the related uncertainty is highlighted in grey. Credit: Elisa Bortolas.

merging binaries, especially at low masses; the concurrent (multimessenger) detection of GWs, electromagnetic radiation, and possibly neutrino emission from the same source will substantially enhance the scientific return of GW observations [484]. Assessing the potential for multimessenger astronomy in the context of LGWA requires a significant step forward in our theoretical understanding of the accretion mechanism in binary systems at different stages of their evolution, besides a more exhaustive comprehension of the localization capabilities once their GW signal is detected.

**Numerical simulations.** A necessary condition for an electromagnetic emission to be produced by the accreting binary is that the host galaxies are sufficiently rich in gas, and gas can efficiently reach the close vicinity of the MBHs. The first condition is likely to be true for LGWA sources, which are most probably embedded in dwarf galaxies at moderate redshifts (see *Electromagnetic emission*).

The probability of electromagnetically spotting an accreting MBH along its orbital decay is also related to the typical timescale the binary spends at each given separation. Keeping in mind the evolutionary path of MBH binaries and the definition of the various “phases” (see section 3.2.2 and the summary provided below), figure 33 qualitatively sketches how much time the binary spends at each distance [524]. The right-most trend at large scales is to be referred to dynamical friction, the central slope is the stellar hardening, and the left-most slope change is to be attributed to the onset of the GW phase. When the two MBHs are still unbound and separated by several kpc, each of them behaves independently of the other. This largest-scale inspiral is likely to take the longest among the pairing phases, so it is intrinsically more likely to spot MBHs at such distances. Furthermore, large
projected separations are easier to spatially resolve for imaging campaigns. Several works have investigated the accretion of MBH pairs in the large-scale, dynamical friction-driven inspiral. In particular, simulations of merging galaxies have shown that the perturbation induced by the merger itself (which renders the two galaxies distorted) can efficiently torque the gas and promote its loss of angular momentum, so that it may reach the vicinity of either MBH and get efficiently accreted [926–928], resulting in variations in the mass ratio of the two MBHs. Accretion feedback released by an inspiralling MBH can evacuate gas from the MBH vicinity, resulting in the so-called *negative dynamical friction*, which would further lengthen the inspiral itself [929] and perhaps enhance our chance to detect binaries in this stage. Further perturbations to the decay timescale that can impact the residence time can be induced by the presence of bars, spirals, star-forming clumps and, in general, morphological distortions that are likely to appear in merging gas-rich systems [526, 527, 930, 931].

As the pairing proceeds below separations of a few pc, the binary gets bound in a Keplerian pair. The transition between the dynamical friction and stellar hardening phase is so prompt — unless the binary mass ratio is very small, $\lesssim 10^{-2}$ [932] — that it is very unlikely to observe the system at this stage (figure 33). The presence of gas in the vicinity of the binary can give rise to electromagnetic signals that would reveal the presence of the binary itself. The gas would settle into a circumbinary disc with an evacuated cavity, carved by the binary, with a size roughly twice the semi-major axis. Under some conditions (namely, if the disc temperature is high enough or if the disc viscosity is large), a significant amount of gas can leak from the cavity’s inner edge forming structures called “mini-discs” around the binary components. The interaction of the binary with the surrounding gas has recently received significant attention as the delicate balance between the positive gravitational torque (adding angular momentum to the binary) provided by the mini-discs and the negative torque of the circumbinary disc (removing angular momentum from the binary) eventually determines whether the binary inspirals or outspirals [529, 933, 934]. Recent numerical works have shown this balance to depend on the disc aspect ratio (i.e. on its temperature) [532, 935]. In more recent simulations at higher resolution, [936] and [937] have shown that the sign of the total torque acting on the binary also depends on the viscosity of the disc, a parameter that is challenging to measure or constrain, as well as whether the simulation incorporates a dynamically evolving binary system.

It is worth noting that, neglecting the effect of the gas, the binary would spend most of its time at the transition between the stellar hardening and GW emission stage, at or below mpc separations (figure 33). As both GW emission and stellar hardening are the least efficient in the transition, this is exactly where gas-driven torques can be the most impactful and affect the orbit, as well as where binaries are more likely to be observed, based on these timescale arguments [938].

Although so far we only mentioned gas accretion, the two MBHs can also reveal themselves as they disrupt a star passing in their vicinity, giving rise to a TDE (section 3.3.3). TDEs can in principle occur at virtually any stage of the inspiral. An enhancement in the TDE rates is likely to occur when the MBHs are still unbound, at the galaxy merger stage [939], and at the binding stage [940, 941]. Furthermore, the Kozai-Lidov mechanism may enhance TDE rates occurring about the secondary MBH even in the hardening stage [942, 943], provided that
the secondary MBH is small enough; since the MBH mass inferred from the TDE light curve
would be much smaller (because corresponding to the smaller BH) than the one estimated
through scaling relations (since the potential of the galactic nucleus is dominated by the
larger BH), this could suggest the presence of a binary system. As our monitoring capabilities
in both optical/UV and X-rays improve, we might expect to be able to select a few binary
candidates by detecting quasi-periodic variability of the TDE-induced accretion flow due to
the orbital motion of the secondary object about the primary. TDEs can also be produced
around the newborn MBH as it receives a GW-induced kick at its formation, even if the
associated TDE rates are expected to be comparable or smaller than those to be expected
around single MBHs [944, 945]. It is important to note that TDEs are not likely to aid the
search for a counterpart of an MBH merger, as the intrinsic TDE rates are too low (at the
very best, one every ~100-1000 years, section 3.3.3) to allow for prompt identification of
the counterpart after the GW event has been observed; yet, they can be fundamental allies
for the broader study of the MBH (binaries) population.

**Electromagnetic emission.** As shown in figure 5, LGWA will be able to detect pairs of
BHs in the last stage of their inspiral and in the merger phases with masses up to about a few
$10^6$ M$_\odot$. The LGWA will confirm the existence of $10^{3-4}$ M$_\odot$ BHs up to $z > 10$ and detect
binaries containing MBHs of $10^5$ M$_\odot$ up to $z \sim 5-6$, while for higher mass ($10^6$ M$_\odot$) LGWA
will be limited to $z < 0.5$. The electromagnetic characterization of MBH/IMBH binaries,
both before and after GW detections, is of utmost importance. This characterization not
only enhances the sky localization accuracy of GW detectors and provides better constraints
on the physical parameters of the binary system, but is also essential for studying the
environments in which these mergers occur [946, 947]. Unfortunately, observations lag behind
theoretical predictions and models. Currently and in the coming years, it will be possible to
provide samples of reliable electromagnetic MBH/IMBH candidates, which can be definitively
confirmed as such only by GW detectors.

*Where to look for electromagnetic candidates for LGWA emitters?* The tight correlations observed between the host-galaxy properties and the BH mass indicate that associated EM signals produced from the more massive BHs detected by LGWA, i.e., upwards of $10^5-10^6$ M$_\odot$, should be mainly searched for in dwarf galaxies. Even more challenging is the case of lower-mass BHs (IMBHs). Indeed, besides some indications from GWs [369, 948], any conclusive electromagnetic evidence for the existence of even a single IMBH is still missing. Among the best candidates for accreting IMBHs, we can include ultra-luminous X-ray sources (ULXs; e.g., [949, 950]). ULXs are defined as point-like, off-nuclear (i.e., sources showing offsets from the dynamical center of their hosts) X-ray sources in nearby galaxies; they are characterized by X-ray luminosity above the Eddington limit for a stellar BH ($L_X>10^{39}$ erg s$^{-1}$). In particular, the still-limited (in number) high-mass end of ULX distribution could be powered by IMBHs. ULXs are largely associated with star-forming regions in spiral and irregular galaxies and possibly blue compact dwarf galaxies; however, cases where they trace the old stellar population in elliptical galaxies are also present (see [951]). Since low-mass BHs in shallow potentials (as, e.g., those of dwarf galaxies) are likely to never settle at the center of their galaxies [952], we may expect that the location of IMBHs is consistent with the offset
position of ULXs. Another astrophysical environment in which low-mass BHs may form is that of globular clusters [923]. As dense stellar clusters evolve, the compact remnants sink to the center of the cluster and segregate in mass. Although old globular clusters are not known for their large gas content, dynamical interactions in the densest regions may convey material (possibly originated by gas lost by stars during their evolutionary phases). If the gas available in such systems is sufficient to fuel efficient accretion, electromagnetic signatures are likely to be expected. Wandering IMBHs in the halo of galaxies are also expected as a result of BH-BH encounters in globular clusters that can eject IMBHs. These objects can also represent one of the consequences of the galaxy merging process if seed BHs were common in the first galaxies that merged to form today's massive galaxies.

*What are the electromagnetic signatures expected for MBHs/IMBHs binaries?* Due to the compactness of binary systems, directly imaging the two BHs requires angular resolutions that are beyond the capabilities of most of the current facilities. The only exceptions are very high-resolution interferometric (VLBI) observations, which allow us to identify double radio cores, if present, down to about parsec/tenths of parsec of relative separation in the local Universe. Observations of individual AGN in dwarf galaxies revealed that they generally show radio jets (e.g. [953–959]). The jet powers are very high (~$10^{42–44}$ erg s⁻¹), with efficiencies >10%, similar to those of more massive galaxies [960]. However, these sources are rare, and dedicated VLBI observations performed on, e.g., radio sources with X-shaped morphology (originally thought to be an expected signature of a binary BH; see [961] and references therein) found very few reliable systems so far [962–964], all with masses larger than ~$10^7$ M☉. The forthcoming/proposed radio facilities, such as the Square Kilometer Array (SKA, [965]) and the Next-Generation Very Large Array (ngVLA, [966]), thanks to their high-resolution, sensitivity, and large-scale survey capabilities are expected to significantly increase the fraction of binary AGN detected at parsec separation in the local Universe [967]. By resolving the sub-microJy population, these facilities will detect the radio emission of IMBHs in the Milky Way globular clusters and will unveil about 60% of wandering BH population in the halo of our Galaxy (if accretion is triggered, for instance, by the encounter with a molecular cloud, [968]).

At closer separations and/or higher redshifts, only indirect methods can be adopted, namely spectroscopic identification and time variability. At sub-pc distances from the accreting BH, the gas, photoionized by the central engine and rapidly moving in its potential well, produces broad emission lines observable in the UV and optical bands. In the presence of a binary system, variable asymmetric/shifted/double-peaked broad emission lines are expected [969–971] owing to the orbital motion of the two BHs. Detailed studies were also performed to explore the potential contribution to the broad line emission from both the circumbinary disc and gaseous streams flowing towards the BH in the presence of a binary MBH [972, 973]. Although the presence of asymmetric/shifted/double-peaked broad emission lines is not ubiquitously associated with a binary (i.e., they can also be produced by the chance superposition of two distinct galaxies and the presence of outflows), these electromagnetic signatures have been used to spectroscopically select several samples of SMBH binary candidates [974–983]. This method works well for unobscured systems with relative separations above the physical distance of the broad emission line regions from the central engine. This translates into the capability of identifying local AGN binary candidates with masses higher than ~$10^{6–7}$ M☉,
i.e., with relative separations larger than about 0.01 pc. Current spectroscopic surveys like the Sloan Digital Sky Survey (SDSS; [984]) are strongly biased against lower-mass AGN and IMBHs; in these plausibly fainter accreting systems, if the broad emission lines are produced, they are difficult to detect and, crucially, proper modeling does not exist yet.

Furthermore, in the presence of a BH binary, periodic modulations are expected in the UV/optical, X-ray, and radio continuum emission, with periods comparable to the binary period or semi-period; we note, however, that also in this case alternative explanations are possible, like jet precession and warped accretion disc of a single MBH. In the presence of binaries, flux modulations can be produced by (a) the periodic feeding of the mini-disc bound to each BH due to the torque exerted onto the circumbinary disc ([985, 986], and references therein), (b) the Doppler boosting associated with the emission produced in the two mini-discs (e.g., [987]), (c) the approaching of two relativistic jets, (d) the effect of a secondary BH crossing the accretion disc of the primary BH. In this regard, the most studied example is the blazar OJ287 (0.1 pc separation at z=0.3056), which was unveiled via a complex modeling of its optical and radio variability and post-Newtonian dynamics (e.g., [988–991]). Besides OJ287, many other promising binary MBH candidates have been discovered through light-curve variations. They have a typical mass higher than $10^7 M_\odot$ and separations even down to milli-pc; their putative periods range from several months to a few years, depending on the mass (e.g., [977, 979–982, 992–999]). Many of these candidates have been questioned; distinguishing real periodicities from stochastic processes acting in a single AGN is still a challenging process (e.g. [1000, 1001]).

In principle, flux modulations could allow us to select lower mass (<$10^7 M_\odot$) binary candidates with relative separations much closer than $10^{-3}$ pc, periods lower than one day, and coalescence time of hours/years (depending on the mass), i.e., MBH/IMBH binaries producing GW detectable by LGWA. However, from the electromagnetic point of view, detecting these systems is still demanding. Due to the decreasing residence time (e.g., [1002] and see figure 33), the number of BH binaries detectable through electromagnetic signatures is expected to decrease going to very low relative separations. This, combined with the expected short periods, implies that, to select significant samples of MBH binaries through photometric periodicity for LGWA, large sky coverage time-domain surveys capable of visiting the same source within a very short period (lower than hours/days) are needed. In addition, high sensitivities are required, as in each observation a signal-to-noise ratio high enough to significantly constrain the presence of periodicity even for lower-mass active BHs is required. This will be possible in the next few years in the optical/IR bands thanks to Euclid [1003], the Vera C. Rubin Observatory [816], and the Nancy Grace Roman Space Telescope [1004] (at least for compact MBHs with masses down to ~$10^5 M_\odot$, [1002]), and in the radio band thanks to the next-generation radio facilities, SKA and ngVLA.

Another electromagnetic signature that can be used to trace the relative motion of the two MBHs down to very close separations is the presence of variable and double-peaked iron emission lines in X-rays; in this case, the centroid energies of the line are Doppler-shifted as a consequence of both the orbital motion and the inclination of the two BHs [997, 1005, 1006]). These iron lines are produced either in the two mini-discs (each BH is expected to be surrounded by its own mini-disc) or in the inner edge of the circumbinary disc. Given the
sensitivity of the current X-ray facilities, the exposure time needed to characterize and trace the variations of the iron emission lines surpasses the very short timescales over which these variations are expected to occur. These are indeed of the order of a few kiloseconds once the masses of the systems that will be mostly detected by LGWA are considered. Another issue that may hamper the detection of variable double-peaked iron emission lines is the limited spectral resolution of the current X-ray detectors, which are unable to ascertain whether a profile is double-peaked or ascribed to a single broad emission line. This calls for dedicated next-generation X-ray telescopes with large collecting area and a few eV spectral resolution (e.g., Athena, [1007]). Still remaining in the field of future X-ray facilities, X-ray detections of binary systems will also take advantage of the increased sensitivity and superb spatial resolution of the envisaged missions AXIS [1008] and Lynx [1009], due to their sharp PSF, of the order of one arcsec, kept almost constant over a large portion of their large field of views (~150–4000 arcmin²), which will allow the detection of the X-ray counterparts of in-spiral and coalescing BHs of ~$10^{5-7}$ M☉ up to z~2.

It is worth reminding that most of the electromagnetic signatures discussed above do not have a unique interpretation in terms of a binary system. However, their joint detection with GWs would remove all possible degeneracies and help constraining the physical and orbital parameters of the pair (see section 3.2.2).

### 3.3.5 Extreme/intermediate mass-ratio inspirals

*Main contributors:* Riccardo Arcodia, Elisa Bortolas, Roberto Della Ceca, Margherita Giustini, Giovanni Miniutti, Paola Severgnini, Cristian Vignali

Binary systems with extreme or intermediate mass ratio are scientifically very interesting, relatively long-lived GW sources. Depending on the mass ratio ($q = M_2/M_1$), they are generally referred to as EMRIs (with $q \le 10^{-4}$) or IMRIs (with $q \approx 10^{-4} - 10^{-2}$). In the presence of gas, these systems (collectively called EMRI hereafter) are also potential sources of electromagnetic radiation, and could therefore represent an important avenue for multi-messenger observations. Here, we discuss in some detail a newly discovered X-ray variability phenomenon that is currently best explained in the context of EMRIs; the so-called X-ray QPEs.

QPEs are soft X-ray flares that repeat quasi-periodically, with observed duration within 0.5–few hours and recurrence every ~2.5–20 hours [1010–1013]. They were so far observed at the centre of galaxies up to $z \sim 0.05$ with stellar masses $\approx 10^{9-10} M_{\odot}$ [1012, 1013], harboring low-mass SMBHs with $M_{BH} \approx 10^{5-7} M_{\odot}$, as inferred from the stellar velocity dispersion [1013, 1014]. These low SMBH mass estimates are supported by the presence of thermal-like soft X-ray emission in the quiescent time between QPEs, likely the Wien tail of a radiatively-efficient accretion disk (with peak temperature $kT \sim 40–70$ eV, [1010–1013]). QPEs are characterised by a sharp flux increase of more than one order of magnitude in the soft X-rays over this quiescence emission, reaching a soft X-ray luminosity of ~$10^{42}–10^{43}$ erg s⁻¹ at their peak [1010–1013]. The spectrum during the QPEs is hotter when brighter with an asymmetric energy-dependence, namely the spectrum on the rise is hotter than that of the decay at the same luminosity level [293, 1013, 1015]. Strikingly, this spectral evolution is seen consistently
across sources with largely different timing behavior in terms of regularity in the bursts
recurrence and amplitude [293, 1010–1013, 1015], suggesting a common physical mechanism.
So far, six sources with recurrent X-ray eruptions were found to follow these observational
properties [1010–1013], with two further candidates showing a similar energy dependence but
only 0.5 – 1.5 flares [1016, 1017]. A harder component, more akin to the hot corona in active
galactic nuclei, is either absent or compatible with background thus unconstrained [1010–
1013]. So far, no simultaneous variability was observed in the optical, UV, IR and radio
wavebands [1010–1013]. However, the multi-wavelength photometry currently available is
likely contaminated or dominated by either the host galaxy emission or that of the quiescence
accretion disk, and therefore can not be unambiguously associated with the X-ray QPEs.

The origin of QPEs is currently still actively debated, although most models identify
QPEs as a high-mass ratio binary [1018–1029]. Some of the latest proposed that QPEs are
the outcome of the interaction between the smaller orbiter and the accretion disk around
the primary massive black hole [1018, 1019, 1026–1029]. For instance, in [1027] the QPE
signal (and the departure from exact periodicity) can be explained considering a geometrically
thick and low mass (~1*M<sub>☉</sub>) disk undergoing Lense-Thirring precession around a spinning
central massive black hole. The disk is regularly pierced by an inspiralling stellar black
hole [1027] or star [1019, 1029] that is misaligned with respect to the plane of the disc. Each
impact between the orbiter and the disk results in an initially optically thick gas cloud
expelled from the disk plane, which adiabatically expands giving rise to the quasi-regular
X-ray eruptions [1027, 1029]. Quite interestingly, some models have proposed that the
compact accretion disk around the primary massive black hole may be the fed via a precursor
TDE [1019, 1021, 1027–1029]. This connection between QPEs and previous multi-wavelength
behavior akin to that of TDEs is supported by observations of the discovery QPE source
GSN 069, such as its UV spectrum [1030] and long-term X-ray evolution [293, 1031], and
by the two additional candidate QPEs [1016, 1017]. If confirmed, this interpretation of the
QPE signal would imply that QPEs are the counterparts of the still-to-be-detected GWs from
EMRIs. The study of those X-ray sources would be fundamental to anticipate, complement
and enhance the study of EMRIs through their gravitational wave emission. For completeness,
it is worth mentioning that alternative models have been proposed for QPEs, suggesting
their origin to be related to disk instabilities [1032–1035].

So far the signal observed from QPEs is restricted to the soft X-ray band ($E < 2$ keV) and the abundance rates of these transients are low ($\approx 0.6 \times 10^{-6}$ Mpc$^{-3}$; [1036]). Therefore, for the purpose of QPE discovery, a soft X-ray telescope with a large effective area and a large field of view is desirable. The WFI onboard Athena may discover $\approx 2$ QPE source per 15-square-degree survey [1036]. WFI will also be able to study spectral and timing properties of QPEs deeper in redshift, up to $z \sim 0.5$ for the most common QPE sources peaking at $\sim 10^{42}$ erg s$^{-1}$ and up to $z \sim 1.3$ for the much rarer ones peaking at $\sim 10^{43}$ erg s$^{-1}$. The large effective area and high spectral resolution of the X-IFU onboard Athena will also allow to assess the presence of ionized outflowing/inflowing gas before, during, and after QPEs.
Acknowledgments

This article has benefited from the program *Lunar Gravitational-Wave Detection* (ICTS / LGWD2023/04) organised by the International Centre for Theoretical Sciences (ICTS), Bengaluru, India.

AS acknowledges the financial support provided under the European Union's H2020 ERC Consolidator Grant "Binary Massive Black Hole Astrophysics" (B Massive, Grant Agreement: 818691). GM is supported by grant No. PID2020-115325GB-C31 funded by MCIN/AEI/10.13039/50110001103. MManc is supported by ERC Starting Grant No. 945155-GWmining, Cariplo Foundation Grant No. 2021-0555, MUR PRIN Grant No. 2022-Z9X4XS, and the ICSC National Research Centre funded by NextGenerationEU. The work of MManc received support from the French government under the France 2030 investment plan, as part of the Initiative d'Excellence d'Aix-Marseille Université — A*MIDEX AMX-22-CEI-02. TP's work is supported by an advanced ERC grant MultiJets. The work of MMagg is supported by the Swiss National Science Foundation (SNSF) grants 200020_191957 and CRSII5_213497; SF, FI and NM are supported by the SNSF grant 200020_191957; EB is supported by the SNSF grant CRSII5_213497. ARP is supported by the Swiss National Science Foundation (SNSF Ambizione grant no. 182044). RA was supported by NASA through the NASA Hubble Fellowship grant \#HST-HF2-51499.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. MWC acknowledges support from the National Science Foundation (NSF) with grant numbers PHY-2010970 and PHY-2117997. AM acknowledges financial support from MUR PRIN Grant No. 2022-Z9X4XS, funded by the European Union — Next Generation EU. CC acknowledges the support of P Sreekumar and the LGWA discussion meeting hosted by the International Centre for Theoretical Sciences (ICTS), Tata Institute of Fundamental Research, India, in April 2023. MAS acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101025436 (project GRACE-BH, PI: Manuel Arca Sedda). MAS acknowledge financial support from the MERAC Foundation. OG is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy — EXC 2121 "Quantum Universe" — 390833306. SCS is funded by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) with funding from the Bundesministerium für Wirtschaft und Klimaschutz under project reference 50OQ2302. RS acknowledges support from the agreement ASI-INAF eXTP Fase B — 2020-3-HH.1-2021 and the INAF-PRIN grant "A systematic Study of the largest reservoir of baryons and metals in the Universe: the circumgalactic medium of galaxies" (No. 1.05.01.85.10). FO acknowledges support from MIUR, PRIN 2020 (grant 2020KB33TP) "Multimessenger astronomy in the Einstein Telescope Era (METE)". PS, RDC, RS, CV, ADR, VB acknowledge a financial contribution from the Bando Ricerca Fondamentale INAF 2022 Large Grant, 'Dual and binary supermassive black holes in the multi-messenger era: from galaxy mergers to gravitational waves'. MG is supported by the "Programa de Atracción de Talento" of the Comunidad de Madrid, grant number 2022-5A/TIC-24235. The work of SM is a part of the (data|theory) Universe-Lab which is supported by the TIFR and the Department of Atomic Energy, Government of India. MZ is funded by the Fonds National
de la Recherche Scientifique (FNRS) under projet de recherche STELLAR (T.0022.22). AB acknowledges support for this project from the European Union's Horizon 2020 research and innovation program under grant agreement No 865932-ERC-SNeX. PA, NVK, PK, RP and VP were supported by the Department of Atomic Energy, Government of India, under Project No. RTI4001. AMB acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 895174. DV acknowledges support from the NASA Hubble Fellowship Program grant HST-HF2-51520. M.T. acknowledges support from MUR through the FARE 2020 scheme, project 'B Massive-LISA', code: R203NMYZJ5.

**Note added.** The main coordinators of the white paper are J. Harms, A. Frigeri, A. Maselli, M. Olivieri, R. Serafinelli, and P. Severgnini. Important section coordination was done by P. Ajith, M. Arca Sedda, R. Arcodia, A. Bobrick, E. Bortolas, MW. Coughlin, J. van Heijningen, F. Iacovelli, V. Korol, M. Mancarella, F. Patat, M. Toscani, and C. Vignali.

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