larger depths, uncertainties become too high and additional data (e.g.
geodetic and electromagnetic sounding data), as well as a priori information,
are included to constrain the velocity profiles. Variations in the inverted
profiles primarily result from differences in the inversion technique and
model parameterization. Beyond depths of $1200\,$km, differences primarily
stem from the type of complementary information used. Over the course of
different Apollo missions, more than 13,000 seismic events were recorded,
although only a subset of them — those with the best signal to noise ratio and
clear P and S identification — were used to infer the lunar internal
structure.

Figure 24: P-wave velocity (left), S-wave velocity (center), and density
(right) profiles from the Moon’s surface to the core (identified in red in the
middle panel) for several seismic velocity models. Data used for generating
these plots are from [41]. Image taken from [10].

In addition to seismic waves, also normal modes of the Moon can provide
information about its internal characteristics. These free oscillations refer
to a standing wave that form when a dynamic system, as for the case of
planetary bodies, is excited. These oscillations manifest as long-period
signals capturing a body response to an excitation. Each normal mode is
characterized by a distinct vibrational pattern and a unique frequency. Then,
the specific set of normal modes for any planetary body is determined by its
size, composition, density distribution, and material properties.
Consequently, studying normal modes can provide valuable information on its
one-dimensional layered structure along with its physical properties. While
normal modes are generally excited by earthquakes on Earth, on the Moon, their
primary source are moonquakes, along with contributions from thermal stress
and tidal forces. Since normal modes are insensitive to the specific
excitation source, they serve as an ideal tool for investigating the Moon’s
interior. This is particularly advantageous because the precise timing and
locations of all meteoroid impacts are not accurately known. [204] inverted
lunar normal modes recorded by the Apollo instruments to gain new insights
into the Moon’s interior. However, Gagnepain-Beyneix et al [201] raised
doubts, demonstrating that the normal modes utilized by [204] fell below the
instrument noise. The high signal-to-noise ratio of the GW detectors and the
advanced noise-cancellation techniques pursued for LGWA will enable noise-free
normal-mode observations, providing a new valuable source of information for
the Moon’s interior. Similar advancements have already been demonstrated for
Mars (see [205]).

Similar to Earth, the main source of data to probe the Moon’s interior is
seismic data. However, the Apollo seismic dataset had limitations due to a
narrow observation band, and stations being located only on the Moon’s near
side. These limitations limit our capability to gain a comprehensive
understanding of the Moon’s internal structure.

There is currently a renewed interest in lunar exploration within the
scientific community, evident in the extensive list of planned lunar missions.
Several of these missions include the deployment of seismographs on the lunar
surface as part of their objectives, with LGWA and its pathfinder mission
Soundcheck being two of them. This development promises to deliver new seismic
data of exceptional quality and broader frequency coverage in the coming
years. As a result, we are on the cusp of a new era in the exploration of the
Moon’s interior, with the potential for significant discoveries and insights
into its inner structure.

One could argue that the number of seismographs will still be too small, thus
limiting the sampling of the Moon’s layers and making inferences more
challenging than for the Earth. A recent paper by [151], however, demonstrates
the feasibility of utilizing even a single seismograph for probing planetary
interiors through global-scale moonquake coda cross-correlations between
seismic events. This methodology has already proven successful in the study of
Mars interior [150], and we propose to apply global-scale waveform cross-
correlations to lunar seismic data as well, taking advantage of the
forthcoming high-quality seismic data. Particularly, the lunar events’ codas
are expected to be significant due to the Moon’s smaller size and relatively
small attenuation compared to Earth. Moreover, the two horizontal components
of ground displacements will allow the application of other techniques aimed
at the Moon interior’s exploration like receiver functions or seismic noise
interferometry [206]. In particular, the latter has been already applied to
the higher frequency ALSEP data [131], making it possible to describe the very
shallow layers, but lowering the frequencies will provide an image of the
Moon’s deeper structure.

####  3.1.3 Moon’s formation history

Main contributors: Alessandra Mastrobuono Battisti, Matteo Di Giovanni

The Moon’s internal structure and its formation history are closely
interconnected. The presence of a distinct core suggests that the Moon was
formed in a hot, violent event, such as a giant impact. The composition of the
mantle and crust aligns with the concept that the Moon was once a molten body
that cooled and differentiated. Insights into the mechanism that formed the
Moon can be inferred from considerations such as the amount of
differentiation, iron content, and the relative size of the core compared to
the other layers. The analysis of the characteristics of seismic waves can
provide information on the energy released during the formation process and
refine the timeline of events that shaped the Moon.

Our understanding of the Moon’s internal structure has been greatly enhanced
by data collected from the Apollo missions, which returned samples of rocks
and soil from the lunar surface. These data, along with seismic observations
and studies of the Moon’s gravitational field, have provided insights on the
Moon’s composition and the arrangement of its different layers.

In this context, LGWA might be the first lunar mission to observe distinct
lunar normal modes excited by moonquakes, which would provide important data
on the deep lunar internal structure. At the same time, site effects on
seismic signals provide information about the Moon’s shallow structure. The
study of the lunar interior with LGWA is outlined in section 3.1.2.

This is important because the Earth-Moon system is unique in several ways,
including the Moon’s size relative to Earth, its small core, and the high
angular momentum of the system. Different theories for the Moon’s origin,
including capture, fission, co-accretion, and a giant impact have been
explored so far [207, 208, 209].

The giant impact theory, first proposed in the mid-1970s [207, 208], is
currently the prominent one, due to its ability to explain several key
features, as shown by [210], [211] and [212]. In this scenario, the Moon forms
as the result of a relatively low-velocity, oblique impact between the proto-
Earth and a planetary embryo of mass similar to Mars, called Theia, occurring
during the latest stages of the formation of the Earth, approximately 4.5
billion years ago [213, 214]. Simulations indicate that, following the impact,
a significant portion ($\geq 60\%$) of the material that forms the Moon
originates from the impactor mantle. Therefore, the composition of the Moon is
expected to mirror that of Theia, which, in principle, should differ from
Earth, as observed with other planets in the solar system. However, analysis
of lunar samples returned from the Apollo missions revealed an exceptional
degree of isotopic similarity between the Earth and the Moon.

Figure 25: A cartoon picture representing the giant impact scenario (top
panel), the multiple impacts scenario (middle panel) and the “synestia” model
(bottom panel). Image credits are provided on the sketches.

The oxygen isotope composition, a proxy for overall Moon composition, shows a
difference $\Delta^{17}O=12\pm 3\,$ppm [215] compared to Earth. This
difference might be even smaller ($1\pm 5\,$ppm), as recently found by [216]
analysing different rock samples. In contrast, Mars has a much larger isotopic
difference, $\Delta^{17}O=321\pm 13\,$ppm [217]. The isotopic composition of
the lunar and terrestrial mantles is indistinguishable also in terms of other
isotopic ratios, including ${}^{50}Ti/^{47}Ti$ [218], and ${}^{182}W/^{184}W$
[219]. This unexpected similarity gave rise to what is referred to as the
“isotopic crisis”, challenging the conventional giant impact theory.

One possible solution to this challenge is that Theia and proto-Earth formed
at similar distances from the Sun, i.e., from material of similar chemical
composition. Various studies have explored and tested this hypothesis by
tracking the composition of Earth and Theia in large sets of solar system
formation simulations. While [220, 221] found this to be a low-probability
event, [222, 223] have demonstrated how late giant impacts frequently occur
between a proto-Earth and a last large impactor of approximately identical
composition, offering a plausible solution to the giant impact challenge. This
finding implies that the resemblance in composition between the Earth and the
Moon may naturally arise as a consequence of a late giant impact.

In addition, several variations of the giant impact scenario have been
proposed to reconcile the theory with observations. Those include a hit-and-
run impact with an even more massive impactor [224, 225], an impact between
two proto-planets of comparable masses [226], impacts with a fast-spinning
proto-Earth [227], multiple impacts [228] and other more complex scenarios
leading to a “synestia”, that is the vaporization of all material involved in
the impact that settles into a torus to then evolve in the Earth-Moon system
[229]) (see Fig. 25999The cartoon image of the Moon formation scenarios is
from https://letstalkscience.ca/educational-resources/backgrounders/origin-
earths-moon for a representation of the different models).

All these scenarios aim to achieve a significant mixing fraction between
proto-Earth and Theia material, intending to address the challenge associated
with the Earth-Moon composition similarity. However, each scenario is
influenced by the need for ad hoc assumptions (e.g. the necessity to quickly
dissipate a large amount of angular momentum) and the inherent low probability
that characterizes them (for instance, the probability of having an impact
between two bodies of the same size is extremely low, see e.g. [223]). Despite
numerous proposals, no individual or combined scenarios fully meet all
Earth–Moon system constraints. Further studies, encompassing a broader range
of parameters and addressing concerns related to angular momentum, isotopic
constraints, and disk masses, are essential to comprehensively understand the
formation of our satellite. However, different scenarios leave different
signatures in the internal structure and composition of the Moon and the
Moon’s internal structure and its formation history are closely
interconnected.

In all scenarios involving an impact to create the Moon, it is believed that
this event generated sufficient heat to form an initially predominantly molten
Moon from the materials involved in the impact. As this magma ocean cooled,
different minerals crystallized and sank or floated to different depths,
depending on their density. This process of fractional crystallization led to
the formation of the Moon’s distinct layers. As mentioned above, seismic data
from LGWA will be instrumental in mapping these layers and understanding their
composition, density variations, and structural characteristics.

Moreover, seismically detected anomalies have indicated the presence of relics
of Theia in the Earth’s mantle, offering constraints on the giant impact
scenario [230]. Seismology of the Moon conducted with LGWA will likely yield
similar results, exploring the lunar mantle and potentially providing future
insights into its structure and the presence of primordial anomalies related
to its formation mechanism. The detection of unexpected structures or
materials will indeed serve as a test for existing theories or will help to
refine them, offering a more comprehensive understanding of the Moon’s
formation and the role of specific impactors.

In conclusion, the formation of the Moon remains an open question, and seismic
data from LGWA will be crucial in exploring the internal structure and
composition of our satellite, providing important constraints on the various
formation scenarios.

####  3.1.4 Geologic processes

Main contributors: Francesco Mazzarini, Goro Komatsu, Luciano Zuccarello,
Alessandro Frigeri

The main geological processes on the Moon are impact cratering and regolith
gardening [231], volcanism [232], mass-wasting/landslides [233], and
tectonics.

The interpretation of composition and morphology of the Moon suggests that
effusive and explosive volcanism took place in the past [232]. Radio-isotopic
dating of first returned lunar samples suggests a volcanism age between 3.1 to
3.8 Gyr ago [234], while recent volcanic rocks sampled by Chang’e 5 show
younger ages, of about 2 Ga [235]. While there is no evidence for active
volcanoes on the surface of the Moon, this does not exclude the presence of
magmatic bodies at depth.

Tectonics on the Moon are studied through charaterization of inactive fault
systems [236] and by investigating any possible recent activity [189]. Impact
cratering and landslides are directly amenable to seismic observations with
LGWA. Landslides are widespread on the Moon and have been generally considered
old in the Lunar geologic record [237, 238]. Studies on recent tectonics [189,
239] are supporting the existence of recent mass wasting processes.

In the following, points 1 to 3 are about the contribution of the LGWA to the
knowledge of the Moon’s interior (see also section 3.1.2) in terms of the
definition of mechanical layering and regolith formation process using the
seismic array of LGWA as ”seismic antenna”. Points 4 to 5 are focused on the
definition and analysis of the main tectonic processes in the area of the
deployment site as well as on the cratering processes in terms of crater
morphology and frequency.

1) The regolith represents the surface where the seismic array will be
deployed in the LGWA experiment. The seismic LGWA array can be used as a
seismic antenna that can investigate the seismic “velocity stratigraphy” of
the site by analyzing the seismic noise [240, 241]. The “velocity
stratigraphy” may provide information on the subsurface layering and the
geometry of the bedrock-regolith interface. Moreover, the analysis of seismic-
acoustic signals (e.g., [242]) of the seismic array in the near side polar
region will likely provide new insights about the lunar seismicity (see also
section 3.1.1) as well as its mechanical stratigraphy providing new clues
about the issues of the cooling of the pristine Moon’s crust [243].

2) Seismic noise cross-correlation applied at pairs of stations unveil
information on the Earth’s crust characteristics and their variations with
time [244]. This noise-based methodology enhances the common features hidden
in the seismic recordings coming from different stations. We would apply this
methodology to implement new protocols and techniques to perform for
retrieving information about the process of the continuous flow of
micrometeorites that govern the resurfacing of the Moon landscape as well as
the production and evolution of regolith.

3) The LGWA array configuration will guarantee us to analyze the kinematic
properties of the seismic wavefield, based on the common waveform model (e.g.,
[241]). In particular, the array methods permit the estimation of the phase
velocity and propagation azimuth of the analyzed signals. The array analysis
will allow researchers to characterize surface waves in terms of dispersion
curves, phase velocity, and their variability with depth. Through the use of a
second array, it will be possible to better constrain the seismic source(s),
especially in terms of depth, increasing the knowledge of the Moon’s interior.
In particular, the detectable frequency band of the seismic antenna depends on
the sensitivity of the sensors and on the maximum inter-distance between the
nodes of the antenna [242].

4) Morphological and structural analysis along with crater counting may be
performed at the LGWA site. The regolith production is mostly due to impacts
and the amount of regolith produced is somehow linked to the size of the
crater (see chapter 2.4.4 Regolith Composition; [106]). Thus, these studies
about the cratering size distribution and craters’ morphology may provide new
data for modeling the production rate of regolith and a possible estimation of
the regolith thickness in the LGWA site and its surroundings.

5) The geological, morphological, and structural studies of the site of the
LGWA will provide information on the deformation style of the area
(compression vs. tension/strike-slip). The spatial distribution and the
patterns of the structures in the area (for instance, the wrinkle ridges, and
grabens) are linked to the mechanical layering of the crust (e.g., [245, 189,
246, 247]). Cooling and tidal despinning of the Moon suggest a tectonic model
inferring compressive tectonics in the equatorial zones, strike-slip tectonics
in the middle latitude belts, and extensional tectonics at the poles [248,
249]; the seismicity detected by the LGWA array may thus be used for better
constraining this tectonic model. Additionally, LGWA observation may provide
information about the complex relationships between tectonism and volcanism,
which are for the moment inferred mostly from the surface geology [250].

###  3.2 Gravitational-wave science

The LGWA will observe GWs at frequencies from 1 mHz up to about 1 Hz, which
makes it the sought-for missing link between the space-based detector LISA and
future terrestrial detectors like the proposed ET and CE. No other GW detector
can observe astrophysical events that involve WDs like tidal disruption events
and SN 1a (sections 3.2.1 and 3.2.2). Only LGWA can observe IMBH binaries in
the early Universe and understand their role in the formation of today’s SMBHs
(see section 3.2.2). Only LGWA can provide early warnings weeks before the
mergers of solar-mass compact binaries formed by NSs and BHs together with
excellent sky localization (section 3.2.3). The observation of DWDs with
precise estimation of their luminosity distance and having identified their
host galaxies with known redshifts will provide a new and independent approach
to measure the Hubble constant (section 3.2.3). Especially LGWA’s unique role
as partner for multiband observations of GW sources together with terrestrial
detectors (or space-based if present) enables a new level of precision in
waveform measurements and searches for imprints of a new fundamental physics
(section 3.2.4).

####  3.2.1 Astrophysical explosions and matter effects on waveforms

Main contributors: Alexey Bobrick, Martina Toscani, David Vartanyan, Tsvi
Piran, Jan Harms, Suvodip Mukherjee, Sourav Roy Chowdhury, Aayushi Doshi,
Suraj N K, Trisha V

##### Stellar mergers –

Stellar mergers are a prominent example where GWs may inform us about physical
processes. The LGWA has the unique capability to observe WD/WD mergers.
Specifying the properties of two white dwarfs, it is currently not yet
established with certainty what specific outcome their merger may produce
[251, 252, 253]. The complexity arises because the nuclear detonation or
deflagration conditions are extremely hard to resolve in simulations [254,
255, 256]. They also have a strong sensitivity to the geometry of the merger
and the composition profiles, and yet, they play a decisive role in the
outcome of the merger [257, 258]. In addition, these processes happen during
the fairly complex gravitational interactions of the two stars [251, 259].
Moreover, following the onset of deflagration or detonation, the exact
nucleosynthesis in the white dwarfs is also challenging to model [260, 261].

There are multiple classes of astrophysical transients and outcomes known to
be produced by double white dwarf mergers [262]. As an example, it is known
that some double white dwarf binaries may detonate fully and lead to bright
supernovae such as normal SNe Type Ia, 91bg-like SNe, SNe Type 91T, or Super-
Chandrasekhar SNe ([263], see also Sec. 3.3.2). Similarly, some white dwarf
binaries may experience a weak detonation and produce Ca-strong SNe, SNe Type
Iax, 08ha-like SNe [263]. In some other binaries, the ignition may be even
more gentle, leaving behind remnants such as another, more massive, white
dwarf, an sdB star or an RCorBor star [251]. LGWA observations may be
transformative for our understanding of such events. Identifying the
connection between the progenitor white dwarf masses and the transient event
they produce would dramatically improve our understanding and modelling
efforts for these systems. Moreover, deci-Hertz GW observations may be the
only way of probing what happened to the binary material in the first seconds
following the detonation, in particular shedding light on the fate of the
surviving object and the physics of explosion. The main caveat in this regard
is the relatively low rate of double white dwarf mergers in the Galaxy and the
local universe.

Less explored avenues involve the physics of mergers of white dwarfs with
neutron stars, stellar black holes [264, 265] and intermediate-mass black
holes [266]. In all these cases, such mergers are expected to produce an
optical transient event and are governed by a complex interplay of nuclear
evolution, magnetohydrodynamics and radiative transfer, especially in the
vicinity of the compact object, which is the dominant feedback source in such
events [267, 268, 269].

Figure 26: Gravitational wave signal (solid blue line) produced as part of the
3-dimensional magnetohydrodynamic simulation of the merger of a 1.4 M⊙ NS with
a 1 M⊙ carbon-oxygen WD companion at arbitrary distance of 50 kpc [270].

Figure 26 represents an example of the evolution of GW strain over the last
$\sim 250$ seconds prior to the merger of a 1.4 M⊙ NS with a 1 M⊙ carbon-
oxygen WD companion, as modeled within the 3-dimensional magnetohydrodynamic
framework [270]. The simulation covers the last 100 seconds of the inspiral
and $\sim$150 seconds after the disruption of the WD. Contrary to the
“chirping” GW amplitude expected for NSNS or BHBH binaries (e.g., [271]), this
example does not exhibit a chirp prior to the merger. However, small
oscillations are noticeable, attributed to the binary’s eccentricity in the
final stages before the merger (Fig. 26, panel a). The GW emission diminishes
abruptly shortly after the tidal disruption of the WD. The authors estimate
that the merger would not significantly contribute to the signal-to-noise
ratio; thus, the inspiral phase will dominate the observed signal [272].

##### Gravitational waves from tidal disruption events –

Tidal Disruption Events [273, 274, 275, 276] occur when a star of mass and
radius ($M_{*},R_{*}$), orbiting a MBH of mass $M_{\bullet}$, is torn apart by
the MBH tides which overcome the stellar self-gravity. After the disruption,
roughly half of the stellar debris is expected to circularize around the MBH
and eventually form an accretion disc. These occurrences lead to
multimessenger emission, including GWs, electromagnetic radiation, and
neutrinos, offering a comprehensive view of the event from the initial
disruption to the later stages (see Sec. 3.3.3 for further details).

In order for a star to be disrupted by the MBH, it needs to be in a region of
the phase-space called loss cone. In terms of angular momentum, the loss cone
is defined as [277]

$\displaystyle L_{\rm lc}\approx\frac{2GM_{\bullet}R_{\rm t}}{\beta},$ (3.1)

where $R_{\rm t}$ is the tidal radius, i.e., the maximum distance from the MBH
to have the disruption of the star

$\displaystyle
R_{t}\approx\left(\frac{M_{\bullet}}{M_{\star}}\right)^{1/3}R_{\star}\approx
2\times 10^{-6}{\,\rm
pc}\left(\frac{R_{*}}{R_{\odot}}\right)\left(\frac{M_{*}}{M_{\odot}}\right)^{-1/3}\left(\frac{M_{\bullet}}{10^{6}\text{M}_{\odot}}\right)^{1/3},$
(3.2)

(see [278] for relativistic corrections to this formula) and $\beta=R_{\rm
t}/R_{\rm p}$ is the penetration factor, that is the ratio between the tidal
radius and the stellar pericenter. In reality, the exact extent of the
disruption depends on the specific characteristics and internal structure of
the star itself [see, e.g., 279]. In general, we can categorize three distinct
cases: when $\beta\gtrsim 1$ the star experiences full disruption; when
$\beta\approx 1$ the star undergoes partial disruption, possibly leading to
quasi-periodic eruptions (e.g., [280, 281]; see Sec.3.3.5, for details); and
when $\beta\lesssim 1$ no disruption of the star occurs.

The main channel responsible for TDEs is two-body relaxation. Two-body
relaxation occurs when a stellar orbit is altered by the gravitational
interaction with another star. In the steep potential of the MBH this
interaction is typically weak, meaning the stellar trajectory is usually not
significantly affected by this deviation (collisionless system). However, in
high-density stellar environments like globular clusters and nuclear stellar
clusters, the cumulative effect of numerous collisions can considerably impact
the star trajectory (collisional system). As a result, a star may have its
pericenter falling within the tidal sphere, increasing the likelihood of a TDE
occurrence.

While so far TDEs have been detected mainly through EM emission, with the
advent of space-based GW observatories we are on the verge of witnessing the
first observations of GW-TDEs [56]. In particular, the most interesting
scenario would be the case of a WD revolving around a light massive BHs
($10^{3}-10^{5}$). This system would in fact produce a GW signal, which could
be a single burst in case of full disruption of the star or a more continue
emission in case of periodic stripping [282, 283]. This emission would present
peak frequencies around the deci-Hz, that is the frequency window where LGWA
is the most sensitive. The detection of such a signal, together with the
electromagnetic and neutrinos counterparts of these sources, could be a
decisive factor for a full characterization of these systems.

Let us focus on the burst emitted during the star disruption, arising from the
time variation of the mass quadrupole of the black hole-star system.
Analytical order-of-magnitude estimates for the peak amplitude and frequency
of this signal read [284, 285]

$\displaystyle h_{\rm gw}$ $\displaystyle\approx 2\times
10^{-22}\beta\times\left(\frac{M_{*}}{\text{M}_{\odot}}\right)^{4/3}\times\left(\frac{R_{*}}{\text{R}_{\odot}}\right)^{-1}\times\left(\frac{M_{\rm
bh}}{10^{6}\text{M}_{\odot}}\right)^{2/3}\times\left(\frac{d}{16\,\text{Mpc}}\right)^{-1},$
(3.3) $\displaystyle f_{\rm gw}$ $\displaystyle\approx\beta^{3/2}\times
10^{-4}\text{Hz}\,\times\left(\frac{M_{*}}{\text{M}_{\odot}}\right)^{1/2}\times\left(\frac{R_{*}}{\text{R}_{\odot}}\right)^{-3/2}.$
(3.4)

From these formulas, we see that a typical scenario involving a Sun-like star
disrupted by a $10^{6}\,\text{M}_{\odot}$ MBH at a distance of 20 Mpc produces
a strain around $10^{-22}$ with frequencies ranging from $10^{-4}\sim
10^{-3}\,\text{Hz}$ depending on how much close the star is to the MBH. For
WD-TDEs instead, considering as typical parameters describing the star a mass
and radius of $M_{*}=0.5\,\text{M}_{\odot},R_{*}=0.01\,\text{R}_{\odot}$, we
expect a similar strain $h\sim 10^{-22}$ but at higher frequencies
$10^{-2}\sim 1\,\text{Hz}$, due to the huge compactness of the WD. Hence, WD-
TDEs emit GWs in a region of the frequency interval where LGWA is the most
sensitive.

To produce GW-TDE waveforms we need to study this signal numerically. Toscani
et al. 2022 [285] have started exploring the huge parameter space describing
this emission, which is described by nine parameters: the MBH mass and spin,
the mass of the star, the penetration factor, the eccentricity and inclination
of the stellar orbit, the viewing angle, the orbital orientation parameter and
the source distance. In their work, Toscani et al. 2022 employed phantom, a
general relativistic smoothed particle hydrodynamics code [286, 287] to
simulate TDEs and calculate the resulting GWs. They have studied how different
parameters affect the signal and started to collect the waveforms in a
catalog. However, to be fully prepared for the first observations of TDEs, it
is crucial to expand this collection of theoretical waveforms by exploring a
wider range of scenarios and parameter combinations.

##### Interactions in binaries –

Interactions in stably transferring binaries are also physically rich, and
involve accretion disc physics, hydrodynamics, possible jet and outflow
formation, and still poorly understood mechanisms of mass loss [288, 289, 290,
291]. With GW observations, it may be possible to constrain the mass transfer
rates in these systems, this way at least indirectly constraining the nature
of interactions [292]. Coupling such observations with optical, X-ray and
radio data may be particularly constraining for understanding the interaction
physics in such binaries [293, 294]. For the Galactic sources, the timescales
in the inner accretion disc match the LGWA band and, therefore, for
sufficiently massive accretion discs, such observations may potentially be
important [295, 296].

Mergers involving bigger objects may appear too slow to produce a measurable
signal in the LGWA band. However, the compact central cores of geometrically
large objects may have much shorter and much more relevant timescales. For
example, common envelope merger events of giants and compact stars that
initially happen on year-long timescales have been recently shown to produce a
measurable GW signal for LISA [297, 298, 269]. Similarly, in mergers of
massive main sequence stars, which take place on tens of minutes timescales,
the geometrically smaller cores may merge on the factor of tens timescales
shorter [299]. Such mergers are common in the Galaxy. For similar reasons,
mergers of white dwarfs with main sequence stars or white dwarfs with the
cores of red giants or stars of the asymptotic giant branch may be visible.

Tidal interactions in highly eccentric systems may also have their unique GW
signature. Such binaries form in hierarchical triple stellar systems [300,
301]. While the effective timescale of the periaston passage may be in tens of
minutes timescale, the tidal oscillations induced in the tidally deformed
stars may have a factor of several higher frequency harmonics. Potentially
detecting such signals will be informative both of the tidal interactions and
the triple stellar dynamics for systems containing tidally interacting stars.

##### Gravitational waves from helium flashes –

Another unique avenue is the detection of GW signature from the helium flash
at the tip of the red-giant branch. Ignition of helium is expected to be off-
center, have a spherically-asymmetric formation and happen on timescales of
tens to hundreds of seconds [302, 303]. Due to the complexity of the onset of
helium flash, little nucleo-hydrodynamic modelling has been done for it. Due
to it happening in the very centres of giants, GWs, along with neutrinos, may
be the only way of observationally constraining these events.

##### Core-collapse supernovae –

In the last two decades, significant efforts have been made in investigating
the GW emission from CCSNe (see [304, 305, 306] for recent reviews).
Asymmetries of the explosion are required to produce GWs. This happens during
three distinct phases: a) immediately at bounce (time-dependent rotational
flattening and post-shock convection, lasting tens of ms); b) a low-state in
which the neutrino-driven turbulence grows (lasting 50–200 ms); and through c)
Proto-NS modal oscillations [307, 308, 309, 310, 311, 312, 313, 314, 304].
These phases, characterised by GW emission in the $\sim$100–1000 Hz frequency
range, are accompanied by an additional low-frequency ($\lesssim$50 Hz)
signal, the so-called GW memory [315] (see also Sec. 3.2.2). This is generated
by asymmetric neutrino emission and aspherical, large-scale matter ejecta
motions [316, 317, 318, 319, 320, 321, 322]. The neutrino memory signal shows
a more secular time-evolution than the matter component, which is due to the
cumulative time-integral of the anisotropy-weighted neutrino luminosity [323].
Sustained turbulent accretion in more massive progenitors results in higher
neutrino luminosities and generally more anisotropic explosions. As with the
matter component, the neutrino component is more pronounced for delayed
explosions of models with higher compactness [324].

Large-scale SN ejecta asymmetries (where jet-like structures may emerge) are
expected (see e.g. [325, 326, 327, 328, 329]) and also supported
observationally by spectropolarimetry [330, 331, 332, 333, 334, 335, 336], but
their GW emissions are associated with frequencies that are beyond the LGWA
observation band (100–1000 Hz) [337, 338, 339, 340] (see also Fig. 27).

Figure 27: Gravitational-wave spectra $2\sqrt{f}|\tilde{h}(f)|$ for CCSNe with
different stellar masses as a function of time projected against strain noise
curves $\sqrt{S_{n}}$ for various GW detectors (adapted from [306]). The
simulations, extending out to $\sim$6 seconds after the bounce, are unable to
constrain the spectrogram in the region 0.1–1.0 Hz.

While most of the published simulations do not cover the decihertz band (see
for instance [306] and Fig. 27) the linear and nonlinear memory effects can
lead to detectable GW signals below 1 Hz [341, 321]. In this way, even with
LGWA, one would be able to study the GW emission generated by the neutrino
processes [342, 319, 306]. As discussed above, the range $>$25 Hz has signals
from the explosion asymmetries, and the range $<$10 Hz carries signals from
the anisotropy of the neutrino emissions. In principle, these are
distinguishable (though they no doubt overlap) and the polarization and signal
together say something about the various asymmetries of the event and the
orientation of the sources with respect to the line of sight. In this respect,
it is worth noticing that a simultaneous detection of the low-frequency memory
signal by LGWA and the high-frequency burst by terrestrial detectors would
provide a more complete understanding of CCSNe.

Although exploration of the lower-frequency CCSN signals and their
detectability have only started in the recent years, the progress made so far
in extending the simulations to several seconds after the bounce, and the
immediate future prospects of reaching out to tens of seconds, carry the
promise of producing consistent results at low frequencies.

##### Gravitational waves from relativistic jets –

The acceleration of a relativistic jet is another source of gravitational
radiation [343, 344, 345]. The resulting GW signal is a memory type one and
its detailed features depend on the acceleration (and possible deceleration)
of the jet, on its overall duration and orientation relative to the observer
[346, 347, 348]. The overall amplitude of the GW signal is, as expected on
dimensional grounds, of order $GE_{\rm jet}/c^{4}d$, where $E_{\rm jet}$ is
the energy of the jet and $d$ the distance of the source. The frequency is the
smaller of the inverse acceleration time and the jet duration. Natural
candidates for jet-GWs are long GRB jets whose typical energy is of order
$10^{51}$ erg and typical frequency of order tens of deciHz. Short GRB jets
are somewhat weaker and their frequency is higher. As those are more frequent,
it might be easier to detect. Most remarkable are hidden jets that take place
in some powerful supernovae [349]. Those jets choked inside supernovae and are
difficult to observe directly. Their unique GW signature might enable us to
confirm their existence. The energies of choked supernovae jets are comparable
to those of long GRB jets [349]. As supernovae are much more frequent than
GRBs it might be expected to detect these signals first [348]. In particular,
the GWs from a powerful hidden jet in a supernovae are likely stronger than
the GWs from the collapse event itself.

####  3.2.2 Populations and formation channels of GW sources

Main contributors: Elisa Bortolas, Manuel Arca Sedda, Pau Amaro Seoane,
Stefano Benetti, Marica Branchesi, Adam Burrow, Enrico Cappellaro, Roberto
Della Ceca, Ines Francesca Giudice, Elisabeth-Adelheid Keppler, Chiaki
Kobayashi, Valeriya Korol, Francesco Longo, Ferdinando Patat, Sourav Roy
Chowdhury, Roberto Serafinelli, Paola Severgnini, Martina Toscani, David
Vartanyan

This section describes the astrophysical sources that could generate a GW
signal accessible to the LGWA. We provide an overview of the main formation
channels for each source and other astrophysical aspects crucial for LGWA’s
source detection and interpretation. We start by detailing the astrophysics of
compact stellar binaries, which can be stellar black holes, neutron stars,
white dwarfs and any combination thereof. We then describe massive BHs and
I/EMRIs: for the former, both bodies in the binary have a mass at least
$>100M_{\odot}$; for the latter, the mass of the secondary body is typically
$<10^{-2}$ times lighter than the primary. We then describe further possible
LGWA sources as stellar tidal disruption events, asymmetric neutron stars and
finally, GWs from type II supernovae via the memory effect. Table 3 summarizes
the main LGWA sources with their properties. The detection horizons for equal
mass binaries and I/EMRIs are shown in Fig. 5.

Source | Formation channel(s)/cause |  $M_{\rm tot}$ [$M_{\odot}$] | $q$
---|---|---|---
stellar binaries | stellar binary evolution, dynamical interactions | 1–102 | $0.1-1$
MBH binaries | Galaxy-galaxy mergers | $10^{3}-10^{6}$ | $0.01-1$
IMRIs | Two-body relaxation, AGN disks | $10^{2}-10^{5}$ | $10^{-4}-10^{-2}$
TDEs | Two-body/resonant relaxation | $10^{3}-10^{6}$ | $10^{-5}-10^{-3}$
Asymmetric NSs | non axisymmetric rotating NS |  $1-4$? | —
Table 3: Here we list the main GW sources for LGWA with their typical
formation channels, together with the total mass, mass ratio accessible to the
LGWA for each of them.

##### Compact-binary mergers of stellar-mass compact objects –

Compact binary systems, comprising WDs, NSs and BHs populate our local
Universe. Approximately $10^{6}$ yr before their merger, these systems emerge
as prime targets for space-based GW observatories. Missions like LISA [350],
TianQin [351], and Taiji [352], set to operate at milli-Hz frequencies, are
anticipated to be functional in the 2030s. Lunar-based projects, such as LGWA
(operating in the deci-Hz regime), although capable of producing exciting
science by themselves, can complement space-based missions, especially at the
high-frequency end of their sensitivity window, effectively bridging the gap
with terrestrial detectors. Furthermore, LGWA can enhance the scientific scope
of compact binaries to approximately 1 Hz, enabling the detection of events
like DWD and NS-WD mergers [10]. Notably, some of these mergers are not
accessible to the terrestrial detectors nor to the lower-frequency space
missions.

Stellar black-hole binaries – The detection of GWs by the LIGO-Virgo-Kagra
(LVK) collaboration, led by the discovery of GW150914 [1], provided proof of
BHs, enriching our knowledge of known merging BHs up to the present population
of nearly 100 objects with masses above 30 M⊙. This was well above the limit
set by previous observations, supporting the evidence for the existence of
binary BHs, and for the formation of new BHs as by-product of a BH mergers.
Such a catalogue of events [55] allows placing constraints on the cosmic
population of merging BHs. Among all observed mergers, there are several
intriguing sources with properties that clearly deviate from the overall
distributions. One of them is GW190521 [353, 354], the first GW source to
leave behind an IMBH with a mass of $\sim 150M_{\odot}$.

With a primary mass of $\sim 85$ M⊙, GW190521 was also a challenging source to
interpret. Stellar evolution models predict a gap in the BH mass spectrum
roughly between $40-300$ M⊙ owing to the onset of pair instability and PPISN
in their progenitors [355]. Stars $>\kern-11.99998pt\lower
4.73611pt\hbox{$\sim$}~{}300M_{\odot}$ may collapse to BHs (see [356] for a
review). The actual range of masses affected by pair instability and PPISN is
highly uncertain, and it depends on the details in stellar evolution models.
Also, star collisions and binary interactions can change the stellar structure
such that the star avoids the pair instability and PPISN and undergoes direct
collapse, possibly leaving a BH remnant with a mass in the gap. BHs in the gap
could also originate from previous BH mergers, following the so-called
hierarchical merger scenario.

It is important to understand the origin of merging BHs to interpret GW
observations. From a theoretical standpoint, the main formation channels for
compact binary mergers are via binary stellar evolution [357, 358, 359, 360,
361, 362, 363, 364, 365, 366, 367, 368, 369], dynamical interactions in star
clusters [370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382,
383, 384, 385, 386, 387, 388, 389, 390, 391, 392], and primordial BH
interactions [393, 394, 395, 396, 397, 398, 399, 400, 401, 402].

Identifying the imprint of different formation channels is not trivial, owing
to the many processes, and related uncertainties that can affect the merging
BH population (see for example [403, 404, 405, 406, 389]). The cosmic star
formation, the metallicity distribution, the star-cluster formation and
evolution, the natal BH mass, spin, kicks, and the parameters of BH binaries
are only a few of such processes and quantities that can dramatically affect
the properties of observed merging BHs [389]. However, only a handful of
parameters, like the binary eccentricity, the mass ratio, and the effective
spin distribution, might be the key to understanding such population of
merging objects.

Among all, the binary eccentricity is considered to be a quantity that can
clearly tell whether a binary merger is formed dynamically or not [407, 78,
79, 408]. Several processes, like common envelope dynamics, mass transfer, and
GW emission, are expected to dampen binary eccentricities and lead to almost
circular orbits ($e<10^{-3}$) even before the binary emits GWs in the
frequency range of terrestrial detectors ($>10$ Hz) (see e.g. [407, 409]).
However, SN kicks or common envelope physics may lead to moderately eccentric
binaries emitting high-frequency GWs [410, 411]. In dynamical environments,
instead, binaries are assembled via strong dynamical interactions, which can
naturally lead to the formation of highly eccentric binaries [412, 413, 414,
415, 416, 417, 418, 388], depending on the environment in which they develop.
For example, eccentric binaries can form via GW captures, secular processes,
and binary–single encounters in quiescent and active galactic nuclei [412,
419, 420, 421], or via three- and four-body interactions in young and globular
clusters [416, 370, 417, 413, 388]. For dynamical mergers, the environment can
also affect the critical value of the binary separation below which GW
emission becomes the dominant evolutionary process and the binary decouples
from the overall dynamics. This means that an eccentric merging binary forming
in a young massive cluster will likely be circularised by the time it reaches
the 1–10 Hz band, but can appear eccentric at lower frequencies observable
with LGWA, whilst a binary forming in a massive and dense galactic nucleus may
form with a sufficiently small separation to be detectable as eccentric source
even with high-frequency detectors (e.g. [422, 408, 417]). Despite some of the
detected GW sources could be eccentric [423, 424, 425, 426, 427], the effects
of eccentricity on detected signals are very similar to those induced by
binary precession, thus making it hard to place stringent constraints on the
eccentricity [428, 429]. Advances in data analysis and waveform templates are
recently enabling the premises to measure the eccentricity of binaries
provided that it exceeds O$(10^{-2})$ [430, 431, 432, 433, 434, 435, 436].
With the possibility to follow the binary over many orbits, lower frequency
detectors may be the key to untangle the two effects and robustly determine
the binary eccentricity [342, 437], helping us to pin down the origin of at
least some sub-population of observed sources [78, 79]. In these regards,
multiband observations with different detectors can help to increase the
accuracy of measured parameters. Multiband detections can be sub-threshold in
LGWA becoming significant only with the help of the other GW detectors
involved in the observation.

Another interesting, possibly distinctive, feature of merging BHs is the
distribution of spins. Ideally, a merging BH binary formed from isolated
binary stellar evolution is expected to lead to two BHs with aligned spins
[438], although some level of misalignment can be induced during the BH
formation process, e.g. during the supernova phase [439, 438, 440, 441, 442,
443]. Conversely, dynamical BH mergers are expected to have spins
isotropically distributed in space, owing to the chaotic nature of the
interactions that drive their formation [409]. This suggests that a precise
measurement of the BH spins and their mutual orientation would help
identifying the binary formation origin. Spins in current GW data analysis are
poorly constrained, and generally the spin orientation of merging binaries is
inferred from the so-called effective spin parameter $\chi_{\rm eff}$ [444],
which is a projection of the spins onto the binary angular momentum, weighted
with the binary component masses. A population of mergers with aligned spins
would be characterised by a $\chi_{\rm eff}$ distribution clearly peaked
around a positive value, whilst such a peak would be centered on zero for
mergers with isotropically distributed spins.

Finally, identifying the location of the merger would definitely help to pin
down the formation scenario. Combining observations performed with different
detectors can significantly help reduce the localisation area, possibly making
it easier to assess the merger birth site (e.g. the properties of host
galaxies and the location within the galaxies) at least on a statistical
basis.

The LGWA can play a crucial role in measuring the aforementioned quantities.
Working in the deci-Hz frequency range, depending on its final sensitivity it
could follow merging BHs for months and weeks before the merger, ensuring high
precision measurements of the masses, the spins, and possibly the eccentricity
evolution of the binary (see Sec. 2.2.3). These are fundamental for further
understanding the evolution of (single) massive stars, and the formation and
evolution of massive star binaries.

White dwarf and neutron star binaries – Mergers of WDs and NSs are extremely
important for understanding the origin of elements [445]. NS mergers are
proposed for one of the astrophysical sites of the rapid neutron capture
process [446], which has been confirmed by the detection of GW and EM
transient ‘kilonova’ [447]. WDs mergers are thought to become sub-
Chandrasekhar-mass Type Ia supernovae [448, 449], which are responsible for at
least some of the Fe production particularly in dwarf galaxies [450]. Compact
stellar systems composed of WDs and NSs are guaranteed GW sources, as some of
them have been already discovered with electromagnetic (EM) telescopes (see
also Section 3.3.1). Some of the shortest period and better characterised of
the known binaries have been also proposed for testing space-based GW
detectors to maximise their scientific output (e.g. [451, 452, 453, 454]).
Some of such systems, known in literature as verification binaries, can be
detected by LGWA [10], and offer the exciting opportunity to study the
astrophysics of compact sources using both their GW and EM emission (e.g.
[455, 456, 457, 458, 459]).

From a theoretical perspective, compact binaries are expected to be several
orders of magnitude more abundant than currently observed by EM telescopes
(e.g. [460, 461]), with binaries composed of two WDs being the most abundant
in the Milky Way and its immediate neighbourhood. Their formation involves up
to several mass transfer episodes; at least one of these episodes has to be
unstable leading to a common envelope, a phase during which the binary can
shrink dramatically [462, 449, 463, 464, 465]. Even though this is one of the
least understood phases in binary evolution, the fact that we do observe close
double compact objects and mergers is a strong indication that something like
a common envelope phase happens [466]. Based on these observations, several
theoretical modelling efforts are being made (e.g. [467, 468]). Thus, the
demographics of compact binaries observed via GW radiation will provide
opportunities to learn new physics and answer key scientific questions related
to the formation and evolutionary processes of close binary systems. Besides
the common envelope physics, this includes questions related to the stability
and efficiency of mass and angular momentum transfer, tides, and accretion
onto compact objects, as well as details of their destruction in supernova
explosions [for a review see 469, 470].

At frequencies $<1\,$mHz, binary population synthesis simulations predict
${\cal O}(10^{7})$ double WDs, followed by ${\cal O}(10^{5})$ NSWDs in the
Milky Way alone [469]. Studying these binary populations with EM facilities
has proven to be technically challenging due to the compact size of these
binaries and WD/NS stars themselves, and so GW detectors – characterised by
different selection effects – are ideally suited for discovering and
characterising them in bulk [471, 472]. Importantly, GW selection effects
enable the study of EM-dim double compact objects at unprecedentedly large
distances throughout the Milky Way, including its most massive satellites and
possibly reaching out as far as the Andromeda galaxy, which otherwise
impossible with EM telescopes [473, 474, 475, 476].

Supernovae Type Ia as probe of white-dwarf binaries – SNe Ia are believed to
originate from WDs which, due to mass transfer or generally, interactions in
close binary systems, grow above the Chandrasekhar limit ($M_{Ch}\sim
1.4\,M_{\odot}$) and are incinerated by a thermonuclear explosion under
degenerate matter conditions. This idea dates back to the early ’70 [477] but
the actual nature of the secondary star remains undetermined. Two alternative
scenarios have been proposed: the donor is a hydrogen or helium main sequence
or a giant star (the single degenerate scenario, or SD), or another WD (the
double degenerate scenario, or DD), which would lose orbital momentum via GW
radiation and eventually merge. In the case of SD systems, mass transfer onto
the accreting WD occurs when the secondary donor star fills its Roche lobe.
This sets a lower limit for the separation between the two stars and the
orbital period (of the order of 10 minutes for the most compact, non-
degenerate donors [478]). In this configuration, the GW emission is weak and
does not significantly affect the binary evolution. On the contrary, in DD
systems the orbital radius naturally shrinks as an effect of angular momentum
loss via GW emission [449]. Because of the small tidal radius of WDs, this
allows a minimum orbital period of tens of seconds (see below). Since the
intensity of the GW emission increases with increasing frequency, this is also
expected to produce a much stronger signal, falling in the 0.01–0.1 Hz domain,
right in the frequency range where LGWA’s sensitivity is at its best. This
difference in orbital periods creates a clear-cut case, in which GW detection
can act as an effective discriminant between the two channels. Using Kepler’s
third law relating the period $P$ and the semi-major orbital axis $a$, and
considering that the GW frequency is given by $f_{GW}=2/P$, it can be readily
shown (see for instance [479]) that

$f_{GW}=\frac{\sqrt{G}}{\pi}(M_{1}+M_{2})^{1/2}\;a^{-3/2}\simeq
0.1\textrm{Hz}\;\left(\frac{M_{1}+M_{2}}{2M_{\odot}}\right)^{1/2}\;\left(\frac{a}{0.02R_{\odot}}\right)^{-3/2},$
(3.5)

where $M_{1}$ and $M_{2}$ are the masses of the two stars. Given the total
mass constraint ($M_{1}+M_{2}\gtrsim~{}M_{Ch}$), the typical WD radius
($R_{WD}\sim 0.01\,R_{\odot}$; [480]) and the minimum WD separation ($a\sim
3R_{WD}$; [481]), Eq. (3.5) yields a maximum GW frequency $f_{GW,m}\sim$ 0.1
Hz when the merger occurs (see also [251], Fig. 21 therein). The time it takes
for a binary system to merge, under the simplifying assumption of a circular
orbit and neglecting possible mass transfer, hydrodynamical and radiative
effects was computed by Peters [482] (see Eqs. (5.9)-(5.10) therein), and can
be expressed as:

$t_{m}=\frac{5}{256}\frac{c^{5}}{G^{3}}\frac{a^{4}}{M_{1}M_{2}(M_{1}+M_{2})},$

which, taking into account Equation 3.5, can be rewritten as:

$t_{m}(M_{c},f_{WG})=\frac{5}{256}\left(\frac{1}{\pi}\right)^{8/3}\;c^{5}\;G^{-5/3}\;M_{c}^{-5/3}\;f_{GW}^{-8/3}\simeq
9.3\;\textrm{yr}\;\left(\frac{M_{c}}{M_{\odot}}\right)^{-5/3}\left(\frac{f_{GW}}{0.1\textrm{Hz}}\right)^{-8/3}$
(3.6)

where $M_{c}=(M_{1}+M_{2})^{-1/5}(M_{1}M_{2})^{3/5}$ is the chirp mass [479].

The SN Ia rate in the Galaxy is well known from direct observations in the
local universe: $r_{\rm Ia}$=(5.4$\pm$1.2)$\times$10-3 yr-1 [483]. This
corresponds to an event every $\sim$200 yr, which is definitely too low to
provide a significant chance of detection of a WD binary merger during the
lifetime of a GW campaign. However, in order to understand the progenitor of
SN Ia, statistical approach can be followed. If the bulk of SNe Ia comes from
DD systems, there needs to be enough such systems with the suitable properties
(mass, merging time) to account for the observed SN Ia rate. Let us assume
that $n(M_{\rm c},f_{\rm GW})$ is the probability density function describing
the distribution of DD systems. Let us then also indicate with $M_{\rm c,l}$
and $M_{\rm c,h}$ the lower and upper chirp mass limits for having a total
mass $M_{1}+M_{2}\gtrsim M_{\rm Ch}$. These limits are set considering the
maximum and minimum mass of a C-O WD: $\sim 1.0M_{\odot}$ [484] and $\sim
0.3M_{\odot}$ [485] ($M_{1}=0.4M_{\odot}$, $M_{2}=1.0M_{\odot}$ and
$M_{1}=M_{2}=1M_{\odot}$, corresponding to an approximate chirp mass range
0.5-0.9 $M_{\odot}$). Finally, let us indicate with $f_{\rm GW,l}$ the lower
limit of the frequency range covered by the GW detector, which corresponds to
the maximum merging time $t_{\rm m,h}$ and maximum orbital period $P_{h}$.
With these settings, the total expected number of suitable DD systems found by
the GW detector is:

$N_{*}=\int_{M_{\rm c,l}}^{M_{\rm c,h}}\int_{f_{\rm GW,l}}^{f_{\rm
GW,m}}n(M_{c},f_{\rm GW})\;dM_{c}\;df_{\rm GW},$

where, as we have shown above, $f_{GW,m}\sim$0.1 Hz. Since, by construction,
all these systems will have merged (and hence exploded) by $t=t_{m,h}$, the
corresponding explosion rate is simply given by

$r_{Ia}=\frac{N_{*}}{t_{m,h}}$

Since $r_{Ia}$ is well known (see above), the argument can be reversed and one
can predict $N_{*}$. For illustration purposes, using this relation and Eq.
(3.6), we have compiled Table 4 for some representative values of $P_{h}$. For
the sake of simplicity, we have assumed $M_{1}=M_{2}=0.7M_{\odot}$
($M_{c}\simeq 0.6M_{\odot}$). We note that the above calculations are a good
approximation only for $P\gtrsim 100$ s. For lower periods (50-100 s,
depending on the WD masses) Roche lobe overflow ensues. This leads to a merger
timescale which is much shorter than what is predicted by Eq. (3.6).

$P_{\rm h}$ |  $f_{\rm GW,l}$ |  $t_{\rm m,h}$ |  $N_{*}$
---|---|---|---
(s) |  (Hz) |  (yr) |
40000 |  5.0$\times$10-5 |  1.3$\times$1010 |  (7.1$\pm$1.6)$\times$107
3600 |  5.6$\times$10-4 |  2.2$\times$107 |  (1.2$\pm$0.3)$\times$105
2000 |  10-3 |  4.6$\times$106 |  (2.4$\pm$0.6)$\times$104
200 |  10-2 |  9.9$\times$103 |  52$\pm$12
Table 4:  GW frequency, merging time and number of DD systems with the
suitable properties required to reproduce the observed Galactic SN Ia rate for
some example values of the maximum orbital period. The largest period
($\sim$11 hours) corresponds to a merging time equal to the Hubble time.

As shown in Table 4, for $f_{\rm GW,l}$=0.01 Hz, which corresponds to a
maximum merging time $t_{\rm m,h}\sim 10^{4}$ yr, $N_{*}=52\pm 12$. Therefore,
to match the observed SN Ia rate, at any given time the Galaxy needs to host
40-60 double-WDs with $M_{1}+M_{2}\gtrsim M_{\rm Ch}$ orbiting with a period
of $\lesssim$200 s. If the number of DD systems found in the frequency range
0.01–0.1 Hz is not enough to account for the observed rate, either SD
contributes in a significant way to the census of progenitor systems, or SN Ia
can originate also from smaller mass systems. These have longer merging times
and hence one would need an even higher number of systems in the considered
frequency range. On the other hand, they may remain undetected because the
intensity of their GW emission is lower than for more massive systems. We
notice that whereas the sub-Chandrasekhar scenario was not the favoured option
to explain the majority of SN Ia [486], in recent years it gained support also
in accounting for the observed SN Ia diversity, and it is considered as one of
the possible scenarios [256, 487]. The simplified approach outlined above is
meant only to illustrate the basic concept. In order to achieve an accurate
prediction of the SN rate to be compared with the observed value, one needs
also to consider the actual distribution of WD masses, the GW strain at a
given frequency and the spatial distribution of sources.

The calculations presented above refer to possible detections within our own
Galaxy. However, Fig. 7 shows a detection horizon for DWD binaries enabling us
to detect DWD systems on the verge of merging in nearby galaxies. Even if the
actual horizon limit depends on the details of the merging process, which is
complex and not fully understood, it is possible to estimate a lower limit on
the rate of DWD mergers detectable by LGWA from the local SN Ia rate, in the
assumption that SN Ia originate from DWD binaries.

Considering the SNs observed by ZTF and ATLAS from 2018 to 2023, we estimate
the number per year of SN Ia within 10, 20 and 100 Mpc by summing the SNe
classified as Type Ia and assuming that, on the basis of percentage of Type Ia
with respect to CCSNe, 40% of unclassified SNe are Type Ia. The numbers
corrected for the effective sky areas observed by ATLAS and ZTF are reported
in the last row of table 5.

|  Maximum distance
---|---
|  10 Mpc |  30 Mpc |  100 Mpc
Observed SN Ia |  $1\pm 0.4$ |  $17.3\pm 1.7$ |  $86.7\pm 3.8$
Observed Unclassified SN |  $8.5\pm 1.2$ |  $31\pm 2.3$ |  $112\pm 4.3$
Corrected SN Ia |  $6.8\pm 1.4$ |  $46\pm 4.0$ |  $200\pm 8.5$
Table 5: The table shows the number of SNe Ia per year. The first row gives
the number per year of SN Ia observed mainly by the ZTF and ATLAS surveys
considering the observations from 2018 to 2023. The second row gives the rate
of unclassified SNe coming from the same surveys. The last row gives the
expected rate of SNe Ia observed by an all-sky monitor. This rate is obtained
by adding the number of observed SN Ia (first row) to 40% of unclassified SNe
(second row), where 40% is the percentage of SNe Ia among classified SN. This
value is then corrected by the surveyed effective areas of about 65%.

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 160 DWD mergers possibly
detectable by LGWA over the 10 years of mission lifetime. This number assumes
a conservative 50% percent duty cycle for optical surveys (due to observation
only during darkness). It does not take into account that some SN Ia are not
detected in the field of view of ZTF/ATLAS due to obstruction of the EM
radiation.

This detection rate enables 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 distance for cosmology.

##### Massive Black Hole Binaries and extreme/intermediate mass ratio
inspirals –

Massive BHs inhabit the nucleus of many galaxies [488, 489], 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 [490]. 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 [491, 492]; 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 [493, 494].
Furthermore, MBHs can be generated through runaway collisions of young stars
formed into a very dense star cluster [495] or through repeated mergers of
smaller stellar black holes again within very dense stellar environments [496,
497]. 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 [498]; (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 ($M_{\rm BH}$–$\sigma$ relation, [488, 489]). 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 $\Lambda$-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
$\ll 1$ pc, GWs drive their final coalescence [499]. 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 $M_{\rm BH}$–$\sigma$ 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 [500], even if large-scale
torquing sources such as bars, spirals, massive clumps and other
irregularities may randomize the inspiral timescale [501, 502]. 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 [503], 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. [504, 505, 506, 507]). 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 [508]. The dynamical evolution of the triple system often results
in the merger of at least two of the three involved MBHs [508].

Several studies have assessed the merger rates of MBH binaries as a function
of the mass of the merging bodies and redshift [509, 510] 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 $z\sim 5$ [510]; 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.

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 Fig 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 Sec. 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⊙, the so-
called supermassive stars (SMS), they likely undergo direct collapse owing to
general relativistic instability (SMSs) (see [511] 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 [512] 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 [513, 489]). The few observational constraints about these
elusive objects come from the detection of IMBHs in dwarf galaxies (see e.g.
[514, 515]), the recent observation of GWs from merging BHs whose final
remnants have masses $<200{\rm~{}M}_{\odot}$, like GW190521 [353], as well as
systems for which a stellar tidal disruption occurred, which temporarily
illuminates the MBH’s environment [516, 517, 518, 489, 519]. Further
observations of IMBH candidates in the $10^{3}-10^{4}{\rm~{}M}_{\odot}$ mass
range have been observed in galactic and extragalactic globular clusters, but
all such observations are controversial and widely debated [520, 521, 522,
523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535]. As seen in
Fig 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. [536, 497, 537, 538, 496, 539, 540, 388]). 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 [541, 542, 392].
Such disks may also become vulnerable to fragmentation in certain conditions,
in which case they form stars and subsequent BHs in situ [543, 544, 545, 546].
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. [538, 547]).

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., [548], although see [549]). The motivation for this is to explain
events with particularly high masses, residual evidence of eccentricity [420,
550], spin alignment vs. antialignment [551] and events which have associated
EM counterparts [552, 553], 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 [554], strengthening the
case for IMRI production in these environments. LGWA will provide valuable
information on the earlier stages of these systems, as well 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 [555, 556, 557, 558, 559, 560].
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 Sec. 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. [561]). Relying on the so-called loss-cone theory,
theoretical models predict typical TDE rates between $10^{-6}-10^{-3}$ events
per galaxy per year [562, 563].

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 [564, 565],
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 $\sim 1$ Gyr) appear to be overrepresented by a factor
of a few tens [566, 567].

LGWA is going to predominately observe TDEs occurring about relatively low-
mass MBHs ($<10^{6}M_{\odot}$), likely inhabiting dwarf galaxies or globular
clusters in the local Universe [489]. 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 [568], as well as their innermost stellar
density profile (which remains virtually unresolvable at the scale of a
fraction of the MBH influence radius [569], where most of TDEs are expected to
be generated [570]). Although earlier works have suggested TDE rates would be
larger about low mass MBHs [562, 571], this idea has been recently revised and
it seems instead that rates about low-mass MBHs drop significantly over time
after an initial burst [570, 572]. 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 and eROSITA 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 [573, 574]. The simplest form of this is an equatorial ellipticity
$\varepsilon$, defined as $\varepsilon=(I_{xx}-I_{yy})/I_{zz}$ 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

$h_{0}=\frac{{4\pi^{2}G}}{{c^{4}}}\frac{{\varepsilon I_{zz}f^{2}}}{{d}}\ .$
(3.7)

For neutron stars observed as pulsars, the frequency and frequency derivatives
are measured and the GW amplitude $h_{0}^{spdwn}$ that could account for the
observed kinetic energy loss (i.e for the observed spin-down) can be
calculated as

$h_{0}^{spdwn}=\dfrac{1}{d}\sqrt{\dfrac{5GI}{2c^{3}}\dfrac{|\dot{f}|}{f}}\ .$
(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.

Figure 28: 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.

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 Fig. 28 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 Sec. 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 Fig. 28. Since the maximum ellipticity that a neutron star crust can
sustain before breaking is likely not greater than $10^{-5}$ [575, 576, 577,
578], 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 $\approx 2$ 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 ($\propto f^{2}$
for deformations, from Eq. (3.7), and $\propto f^{3}$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 [579],
the prospects for detection would significantly increase with a sensitivity
$\approx$ 20 times larger. Relative to the results shown in Fig. 28, 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

$\dot{f}=-\frac{2\pi^{2}B_{d}^{2}R^{6}f^{3}}{3Ic^{3}}-\frac{512\pi^{4}G\epsilon^{2}If^{5}}{5c^{5}}\
.$ (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 [580]. 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. [581] 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. [582,
583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597,
598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608]) and completely new
potential observables [596, 609, 610, 611, 612, 613, 614]. 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 $\Lambda$CDM 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 [447] 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 [615, 616].
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.
[617]), 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

$d_{L}(z)=(1+z)\dfrac{c}{H_{0}}\int_{0}^{z}\dfrac{{\rm
d}\tilde{z}}{\sqrt{\Omega_{m,0}(1+\tilde{z})^{3}+\Omega_{r,0}(1+\tilde{z})^{4}+\rho_{\rm
DE}(\tilde{z})/\rho_{0}}}\,,$ (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 $\rho_{\rm
DE}(z)$ 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 [582], 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 [592, 596, 606, 593, 595, 594, 597,
612] and simulations [587, 589, 618, 619, 620, 621, 622, 623, 624]. 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 [625, 599, 590,
601, 626, 627, 628, 629, 605, 630]. This method can further explore weak
gravitational lensing [600, 631, 632] and provide tests of GR modifications at
cosmological scales in a model-independent way [627]. Other methods to employ
GW detections for cosmological purposes include, e.g., exploiting multiple
images of strongly lensed events [633, 634, 635]; having prior knowledge on
the redshift distribution of the observed sources [636, 637]; using tidal
distortions of NSs combined with EoS knowledge to extract information on the
source-frame masses [586, 638, 639]; or measuring the angular barion acoustic
oscillation scale from a high number of well localized events [640].

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 $H_{0}$ [641]. The estimate
was $H_{0}=70\begin{subarray}{c}+12\\\ -8\end{subarray}~{}{\rm
km\,s}^{-1}\,{\rm Mpc}^{-1}$ 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 $H_{0}=70.3\begin{subarray}{c}+5.3\\\
-5.0\end{subarray}~{}{\rm km\,s}^{-1}\,{\rm Mpc}^{-1}$ [642] (see also [643]
where using a different treatment for the peculiar velocity of the host galaxy
it has been estimated $H_{0}=68.3\begin{subarray}{c}+4.6\\\
-4.5\end{subarray}~{}{\rm km\,s}^{-1}\,{\rm Mpc}^{-1}$ and
$H_{0}=68.3\begin{subarray}{c}+12\\\ -8\end{subarray}~{}{\rm km\,s}^{-1}\,{\rm
Mpc}^{-1}$ 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 [592, 593, 594, 595, 596, 597], as
well as other modified gravity observables [596, 610, 614, 611]. 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
$H_{0}=75\begin{subarray}{c}+40\\\ -32\end{subarray}~{}{\rm km\,s}^{-1}\,{\rm
Mpc}^{-1}$ [593].

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 $\gtrsim{\cal O}({\rm
kHz})$ 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 Sec. 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 ${\cal O}({\rm few})$
forecasted BNS multimessenger detections per year at LGWA (see Sec. 2.2.1).

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 localisation area thanks
to the long time spent by the sources in the dHz detector band, and thus the
huge effective baseline (see Sec. 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 $\propto 1/\sqrt{N_{\rm events}}$.
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 Sec. 3.3.4).
Finally, as shown in Fig. 7, LGWA will be able to detect coalescences of DWD
and NSWD binaries up to $\sim\\!100~{}{\rm Mpc}$. 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 Sec.
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. [644, 645, 646] 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. The characteristic frequency $f_{0}$ of a CGWB at present time is
determined by the Hubble rate at the time of generation of the background,
redshifted to the present time [645]

$f_{0}=H_{*}\frac{a_{*}}{a_{0}}\simeq 1.65\times
10^{-7}{\rm\,Hz}\,\frac{T_{*}}{{\rm
GeV}}\left(\frac{g_{*}}{100}\right)^{{1}/{6}}\ ,$ (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, $R_{*}H_{*}\leq 1$. Hence, LGWA
allows to explore e.g. phase transitions occurring in the early universe at
temperature scales up to $T_{*}\sim{\cal O}(10^{6})$ GeV.101010Signals sourced
by phase transitions at high temperatures $T_{*}\gtrsim{\cal O}(10)$ GeV may
experience a different redshift, see, e.g., [647].

In Fig. 29 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 [648, 649, 650] 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
[649] 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 [651, 652] 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 $G\mu\sim 10^{-6}[\eta/(10^{16}{\rm
GeV})]^{2}$). In a first-order phase transition (FOPT), GWs can be produced by
the collision of the broken-phase bubble walls [653, 654, 655, 656, 657, 658],
the production of sound waves [659, 660, 661, 662, 663, 664, 665], and the
development of magnetohydrodynamical turbulence in the primordial plasma [666,
667, 668, 669, 670, 671, 672, 673, 674]. In Fig. 29, we show the CGWB produced
by sound waves, which is expected to produce the dominant signal from a FOPT
[675, 676, 652], based on the results of [664] 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,
[677]): 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 $[S_{n}(300~{}{\rm Hz})(300~{}{\rm Hz})^{3}]_{\rm
LIGO-O5}/[S_{n}(0.3~{}{\rm Hz})(0.3~{}{\rm Hz})^{3}]_{\rm LGWA-Si}\approx
5\times 10^{4}$.

Figure 29: Power-law integrated sensitivity curves [normalised to
$h=H_{0}/(100~{}{\rm km\,s}^{-1}{\rm Mpc}^{-1})$] 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 $T_{\rm obs}=1~{}{\rm yr}$ 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 $\xi_{w}=0.4$) [664] and cosmic
strings (with different string tension parameters $G\mu$) [651] observable at
LGWA.

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 SGWB searches. We show in Fig. 29 the power-law
integrated sensitivity curves (introduced in [678]) computed as in App. A of
[65], 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 [679]); 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.111111Notice that, since in Fig. 29 we report a power-law
integrated sensitivity curves for the full LIGO-Virgo-KAGRA network during O5,
we do not observe the same $\sim{\cal O}(10^{4})$ improvement reported in the
text in the case of a single LGWA compared to a single LIGO. From Fig. 29 we
can appreciate how LGWA could, e.g., improve on the LVK results for cosmic
strings [680] 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 [681] of the detectors has a nearly
constant value of 1. Indeed, using the results in [35, 18], it is
straightforward to see that, for two antipodal seismometers, the overlap
reduction function $\gamma(f)$ at frequency $f$ reduces to
$\gamma(f)=15~{}j_{2}(\Phi(f))/\Phi^{2}(f)$, where $\Phi(f)={4\pi fR}/{c}$
(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 [682, 683]. 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 [684, 685].

The ppE approach builds upon the parametrized post-Newtonian formalism [686,
687, 688, 689], and introduces shifts in both the phase and the amplitude of
the waveforms, which can be written in frequency domain as

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

where $u\equiv(\pi{\cal M}f)^{2/3}$, ${\cal M}$ is the binary chirp mass, $f$
is the Fourier frequency, and $(\alpha,\beta,\gamma,\delta)$ are the ppE
parameters [690, 691]. $\beta$ 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 [692, 693, 694].

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 $-1$PN dipolar
emission, that arises in many extensions of GR which include extra fields
coupled to the gravity sector [695, 696]. Such negative pN terms are more
effective at lower frequencies, i.e. when systems orbit far from the
coalescence [697, 698, 699]. 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
[700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711]. 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 [705].

##### 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 [712], exploiting observations of the
inspiral and the post postmerger phases of the coalescence, and studying
different but complementary signatures within the GW spectrum. a

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 [713, 714, 715, 716]. According
to the no-hair theorem [717, 718] of GR, the oscillation frequencies and
damping times of these QNMs can be characterized solely by the mass and spin
of the remnant BH [719, 720, 721]. Thus, detecting multiple modes will provide
crucial information about the nature of the remnant, and enable interesting
tests of GR [722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734].
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 [735, 736, 737, 738, 739,
740, 741, 742, 743, 744, 745, 746, 747]. 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 [748, 749, 750].

Spin-induced multipole moments – Multipole moments provide yet another set of
observables which can be exploited to test the nature of compact objects [751,
752, 753, 754, 755]. 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 [751]. 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 [756, 712, 757]. 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 [701, 703, 702, 700].
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 [758, 708,
759, 760, 761, 762]. 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 [708]. These dense DM
spikes may arise from the adiabatic growth of astrophysical BHs at the centres
of DM halos [763, 764, 765]. Alternatively, if PBHs are produced in the early
Universe [766], they will naturally be surrounded by large overdensities of
particle DM (assuming that the PBHs do not constitute all of the DM
themselves) [767, 768, 769]. PBHs themselves might significantly contribute to
dark matter [770], and they might have acted as seeds ot today’s supermassive
black holes [771], whose early presence in the Universe — especially of those
revealed by the latest JWST discoveries [772] — challenges models of their
formation.

Feedback effects are expected to disrupt the DM spike if the secondary compact
object is too massive [773, 708], 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 [764, 774]. Intermediate-mass BHs
instead are expected to live in more pristine environments [489] 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 $f_{\mathrm{GW}}\sim
0.1-1\,\mathrm{Hz}$ 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 [705, 709,
711]. 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 [775]. 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 [706]. 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., [400]) or could be produced by the
accretion of particle dark matter into neutron stars or white dwarfs (dark
matter transmutation; see, e.g., [776]).

###  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 Sec. 3.3.1. The detection of SNe together with associated GW
signals would be a major breakthrough (Sec. 3.3.2). Rare signals observed in
EM and GW might come from TDEs of WDs near the horizon of BHs (Sec. 3.3.3).
Another multi-messenger study concerns MBH binaries in AGNs, where X-ray
observations can complement the GW detection (Sec. 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 (Sec. 3.3.5).

####  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 (c.f. Sec. 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 [777]. 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 [778]. Given
that WDs represent the natural end stage of low-mass star evolution ($\sim$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 [779]. 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 [471]. 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 $\times 100\,\text{km\,s}^{-1}$, 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 [780] targeted
around 1000 bright WDs, confirming several dozen DWDs [781]. More recently,
[782], 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)
[783] and the Zwicky Transient Facility (ZTF) [784], among others, are
beginning to detect WD binaries regularly, with spectroscopy following for
confirmation and characterization [785]. In addition, DWD binaries can be
found in exoplanet-focused time-domain photometric surveys, as exemplified by
detections from Kepler and TESS space missions [786, 787, 788]. In the near
future, the Gaia survey [789] is poised to offer new insights into the field
[790]. Additionally, state-of-the-art facilities like the Vera C. Rubin
Observatory [791], along with tools specifically tailored for spectroscopic
follow-ups like SOXS [792], promise to significantly contribute to the area.
They are anticipated to yield hundreds, if not thousands, of new
identifications [456, 793]. 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 [794],
modeled as a double degenerate binary with orbital period of 321 s, first
identified analyzing ROSAT data [795]. 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 $\sim 2$
hours. Several tens of these binaries are presently known in this category
[796, 777]. 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) [797].

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 $\sim 6$ %, aligning with predictions from population synthesis [798].
Out of these, merely around $\sim 10$ % [799] 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 [800, 801]. 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 [802] 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 [803]. 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
type Ia supernova progenitors, as reviewed by various authors such as [804,
805, 806, 800, 807]. 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 [808].
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 [809, 810]. 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,
Type Ia supernovae, 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 Fig. 3 of [811]. The case of type Ia SN holds broader historical
significance and particular scientific interest for LGWA, and is therefore
discussed in detail in Sec. 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 [812,
813, 814]. The merger events involving NS or BH with WD have been extensively
studied through various hydrodynamical and nuclear-hydrodynamical simulations.
Some notable references in this research include [815, 265, 816, 268, 270]. It
is generally anticipated that all NSWD mergers result in unstable mass
transfer [817]. 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 [815]. Nuclear burning within the disk progresses
steadily, potentially with a weak detonation, and it results in the production
of small amounts of 56Ni, typically at most 10-2 M⊙ [816]. This limited amount
of 56Ni 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 56Ni. 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; the strategy for LGWA to detect
them is to identify their quasi-monochromatic emission by integrating it over
the whole lifespan of the mission, similarly to what is done for continuous
wave searches by current gravitational wave detectors. The presence of
eccentricity – expected for NSWD binaries if the NS forms after the WD [800] –
gives rise to higher-frequency harmonics of the emission [818, 677], 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 Sec. 2.2.1, deci-Hz
gravitational wave sources in our own Milky Way are very likely to be observed
regardless of their orientation, while sources in nearby, satellite galaxies
might be observed in favorable conditions.

The expected number of multi-messenger detections will probably be modest,
even though our Milky Way hosts an estimated $\mathcal{O}(10^{8})$ DWD and
$\mathcal{O}(10^{7})$ NSWD binaries [469]. This is because only a small
fraction exists as short-period binaries. The rarity of such binaries arises
because the ratio of DWD to single WD steeply decreases with shorter periods
due to the reduced time until merger. For context, when extrapolating from the
Galactic supernova Ia rate (assuming all derive from DWD mergers), it is
projected that the Milky Way contains merely a few tens of DWD binaries with
periods $(P<200\,$s) (GW frequency $(f>0.01\,$Hz) and a few hundreds with
$P<500$ s $f>0.004$ Hz). We note that this is a conservative estimate as not
all DWD mergers necessarily result in type Ia supernovae [811]. Nevertheless,
pinpointing even a limited number of ultra-short period DWD/NSWD binaries can
yield significant astrophysical insights. 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 type Ia supernova explosion,
would be a groundbreaking discovery, much like for the first multi-messenger
observation of binary NS merger GW170817 [819]. The likelihood of observing
such an event within a 10–20 yr observational span remains low, especially
considering the Milky Way’s type Ia supernova rate of 1 per century (see Sec.
3.3.2).

In the event an NS does not collapse into a BH post-merger, Morán-Fraile et al
[270] estimates 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, [820]), 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, 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 [821] for a review).
Despite their importance, the nature of their progenitors is still unclear and
two alternative scenarios have been proposed: the SD scenario and the DD
scenario (see Sec. 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 [822] or absence [823] 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 [824]; iii) the failed search of a precursor
star in pre-explosion archival images [825] 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 [781] 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 [826], although the strength of this argument is
debatable [827].

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 [828]). 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 [829].

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 [814], others allow for a long delay between the