# High-redshift supermassive black hole mergers in simulations with dynamical
friction modelling

Colin DeGraf1, Nianyi Chen2, Yueying Ni3, Tiziana Di Matteo2,4,5, Simeon
Bird6, Michael Tremmel7, Rupert Croft2,4
1 Department of Physics, Truman State University, Kirksville, MO 63501, USA
2 McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon
University, Pittsburgh, PA 15213
3 Center for Astrophysics, Harvard & Smithsonian, Cambridge, MA 02138, US
4 NSF AI Planning Institute for Physics of the Future, Carnegie Mellon
University, Pittsburgh, PA 15213, USA
5 OzGrav-Melbourne, Australian Research Council Centre of Excellence for
Gravitational Wave Discovery
6 University of California, Riverside, Riverside CA 92521, USA
7 Physics Department, University College Cork, Cork, Ireland

###### Abstract

In the near future, projects like LISA and Pulsar Timing Arrays are expected
to detect gravitational waves from mergers between supermassive black holes,
and it is crucial to precisely model the underlying merger populations now to
maximize what we can learn from this new data. Here we characterize expected
high-redshift ($z>2$) black hole mergers using the very large volume Astrid
cosmological simulation, which uses a range of seed masses to probe down to
low-mass BHs, and directly incorporates dynamical friction so as to accurately
model the dynamical processes which bring black holes to the galaxy center
where binary formation and coalescence will occur. The black hole populations
in Astrid include black holes down to $\sim 10^{4.5}M_{\odot}$, and remain
broadly consistent with the TNG simulations at scales $>10^{6}M_{\odot}$ (the
seed mass used in TNG). By resolving lower-mass black holes, the overall
merger rate is $\sim 5\times$ higher than in TNG. However, incorporating
dynamical friction delays mergers compared to a recentering scheme, reducing
the high-z merger rate mass-matched mergers by a factor of $\sim 2\times$. We
also calculate the expected LISA Signal-to-Noise values, and show that the
distribution peaks at high SNR ($>$100), emphasizing the importance of
implementing a seed mass well below LISA’s peak sensitivity ($\sim
10^{6}M_{\odot}$) to resolve the majority of LISA’s GW detections.

##  1 Introduction

Supermassive black holes (SMBHs) have been found to exist at the center of
galaxies (Kormendy & Richstone, 1995), with a strong correlation to host
galaxy properties (Magorrian et al., 1998; Gebhardt et al., 2000; Graham et
al., 2001; Ferrarese, 2002; Tremaine et al., 2002; Häring & Rix, 2004;
Gültekin et al., 2009; McConnell & Ma, 2013; Kormendy & Ho, 2013; Reines &
Volonteri, 2015; Greene et al., 2016; Schutte et al., 2019). These
correlations hold true across cosmic time, suggesting a coevolutionary growth
between black holes and the galaxies which host them. These galaxies (and the
dark matter halos in which they are located) are expected to merge (e.g.
Fakhouri et al., 2010; Rodriguez-Gomez et al., 2015). Following a galaxy
merger, the central black hole from each progenitor galaxy can migrate toward
the galactic center of the newly merged galaxy, where they can form a binary
and eventually merge together themselves (e.g. Mayer et al., 2007). Black hole
mergers produce strong gravitational wave (GW) signals, and the coalescence of
a pair of supermassive black holes found at the center of galaxies will
produce the strongest GW signals in the Universe.

In the past several years, gravitational waves produced by black hole mergers
have been detected using interferometers (e.g. Abbott et al., 2016), but size
limitations due to the ground-based nature of these instruments mean that
detections to this point have been limited to mergers between stellar mass
black holes. Higher-mass mergers (i.e. between SMBHs) produce GWs with much
longer wavelengths, beyond the sensitivity of ground-based interferometers.
However, the upcoming Laser Interferometer Space Antenna (LISA) space mission
will focus on lower-frequency GWs corresponding to higher-mass mergers, with
sensitivities peaking at $\sim 10^{4}-10^{7}M_{\odot}$ (Amaro-Seoane et al.,
2017). Furthermore, Pulsar Timing Arrays should be capable of detecting even
higher mass mergers, reaching black holes above $10^{8}M_{\odot}$ (e.g.
Verbiest et al., 2016; Desvignes et al., 2016; Reardon et al., 2016;
Arzoumanian et al., 2018). The GWs detected by these observations should
provide a new and powerful mechanism to study SMBHs and their connection with
their host galaxies.

SMBH mergers detected through GWs can provide a wide range of constraints on
our understanding of black hole - galaxy coevolution, including estimating the
rate at which SMBHs merge (e.g. Klein et al., 2016; Salcido et al., 2016;
Kelley et al., 2017; Ricarte & Natarajan, 2018; Katz et al., 2020; Volonteri
et al., 2020), the expected merger/coalescence timescale (e.g. Volonteri et
al., 2020; Banks et al., 2022), how mergers influence the scaling relation
between black holes and their host galaxies (e.g. Volonteri & Natarajan, 2009;
Simon & Burke-Spolaor, 2016; Shankar et al., 2016), gas environment and
accretion efficiencies (e.g. Kocsis et al., 2011; Barausse et al., 2014;
Derdzinski et al., 2019), how SMBH seeds initially form (e.g. Sesana et al.,
2007; Ricarte & Natarajan, 2018; DeGraf et al., 2021), and the potential
connection with host galaxy morphologies (with multimessenger studies that
combine GW and electromagnetic information, e.g. Volonteri et al., 2020;
DeGraf et al., 2021). To maximize what we can learn from the initial SMBH
merger detections, it is important to understand the underlying merging
populations.

Cosmological simulations provide an ideal mechanism to characterise merging
populations, as they self-consistently model both black holes and galaxy
formation, encompass large volumes which provide robust statistical samples,
and span a wide redshift range to investigate evolution over cosmic time.
Current simulations (e.g. Vogelsberger et al., 2014; Dubois et al., 2014;
Schaye et al., 2015; Feng et al., 2016; Pillepich et al., 2018a; Henden et
al., 2018; Davé et al., 2019; Chen et al., 2022b) resolve a wide range of
scales, with BHs ranging from $\sim 10^{4}-10^{10}M_{\odot}$, resolving black
hole growth, mergers, and their host galaxy properties. The majority of
cosmological simulations, however, tend to only resolve higher-mass black
holes (on the order of $10^{6}M_{\odot}$), which misses the majority of LISA-
detectable mergers. Additionally, simulations frequently do not physically
model the infall/inspiral of the black holes leading to coalescence, which has
the potential to significantly impact merging black hole populations (see,
e.g. Volonteri et al., 2020; Banks et al., 2022). Here we study the Astrid
simulation (Ni et al., 2022), which includes both low-mass seeds (down to
$10^{4.5}M_{\odot}$), thereby resolving mergers at the peak of LISA’s
sensitivity; and it directly incorporates dynamical friction to model the
orbital dynamics to small scales, providing a more accurate probe of
inspiralling black holes, and prevents mergers from occurring during fly-by
encounters (Chen et al., 2022a).

This paper is organized as follows: in Section 2 we provide an overview of the
Astrid simulation, including black hole and dynamical friction models. In
Section 3 we discuss the overall black hole population in Astrid, as well as
the expected merging populations. In Section 4 we investigate the
gravitational waves emitted by these mergers, and combine with LISA
sensitivity to calculate the expected Signal-to-Noise (SNR) ratios for the
full merger samples (Section 4.1). Finally, we summarize our conclusions in
Section 5.

##  2 Method

In this work, we use Astrid (Bird et al., 2022), a cosmological simulation run
using a version of the MP-Gadget smoothed-particle hydrodynamics (SPH)
simulation code, a highly scalable version of the Gadget-3 code (Springel et
al., 2005). The simulation consists of a (250 $h^{-1}$ Mpc )3 volume
containing $2\times 5500^{3}$ particles, resolving galactic halos down to
$10^{9}M_{\odot}$ from $z=99$ to $z=2$. The cosmological parameters for the
simulation are based on measurements from Planck Collaboration et al. (2020)
($\Omega_{0}=0.3089,\Omega_{\Lambda}=0.6911,\Omega_{b}=0.0486,\sigma_{8}=0.82$,
and $h=0.6774$), and has an initial mass resolution of $M_{DM}=6.74\times
10^{6}\ h^{-1}\ M_{\odot},M_{gas}=1.27\times 10^{6}\ h^{-1}\ M_{\odot}$ and
gravitational softening length of $\epsilon=1.5\ h^{-1}\ kpc$.

The Astrid simulation includes detailed models for galaxy formation and
evolution, including reionization (Battaglia et al., 2013; Faucher-Giguère,
2020) with self-shielding (Rahmati et al., 2013), star formation (Springel &
Hernquist, 2003) with associated feedback (Okamoto et al., 2010) and metal
return (Vogelsberger et al., 2013; Pillepich et al., 2018b). For a more
detailed description of the physics modelled in this simulation, see Ni et al.
(2022); Bird et al. (2022).

Of particular import for this analysis is the implementation of black holes in
Astrid (Ni et al., 2022; Chen et al., 2022a). Black holes in the Astrid
simulation are treated as collisionless sink particles, inserted into halos
above a mass threshold of $M_{\rm{halo}}=5\times 10^{9}\ h^{-1}\ M_{\odot}$
and $M_{*}=2\times 10^{6}\ h^{-1}\ M_{\odot}$ which do not already contain a
black hole particle. Rather than using a fixed seed mass for black holes,
Astrid selects a mass for each newly-seeded black hole from a power law
distribution (power law index $=-1$) from $3\times 10^{4}-3\times 10^{5}\
h^{-1}\ M_{\odot}$, intended to remain broadly consistent with a variety of
SMBH formation pathways and their subsequent growth (e.g. Begelman & Rees,
1978; Madau & Rees, 2001; Volonteri et al., 2003; Bromm & Loeb, 2003; Regan &
Haehnelt, 2009; Katz et al., 2015; DeGraf & Sijacki, 2020). Once seeded, the
black holes grow by merging with other black holes, and via mass accretion
following a model based on a Bondi & Hoyle (1944)-like formalism applied to
the SPH kernel of the black hole (periods of super-Eddington accretion are
permitted, but capped at $2\times$ the Eddington rate). Accreting black holes
are assumed to produce a bolometric luminosity proportional to the accretion
rate (at 10% efficiency), and 5% of the radiated energy is assumed to couple
thermally to the surrounding gas (this feedback energy is deposited
isotropically among gas particles within the SPH kernel). Please see Ni et al.
(2022) for additional details.

Rather than using a respositioning scheme to simply move all black holes
toward nearby potential minima, Astrid implements a dynamical friction model
for black holes (Tremmel et al., 2015; Chen et al., 2022b). We assume a
Maxwellian distribution for the velocity distribution of the surrounding
particles (both stars and dark matter), such that the dynamical friction force
can be calculated (see Binney & Tremaine, 2008) as

$\textbf{F}_{\rm{DF}}=-4\pi\rho_{sph}\left(\frac{GM_{\rm{BH}}}{v_{\rm{BH}}}\right)^{2}{\rm{log}}(\Lambda)\mathcal{F}\left(\frac{v_{\rm{BH}}}{\sigma_{v}}\right)\frac{\textbf{v}_{\rm{BH}}}{v_{\rm{BH}}}\,,$
(1)

where $\rho_{\rm{sph}}$ and $\sigma_{v}$ are the density and velocity
dispersion of the surrounding dark matter and star particles,
$\textbf{v}_{\rm{BH}}$ is the velocity of the black hole relative to the
surrounding medium, $\Lambda$ is the Coulomb logarithm

$\Lambda=\frac{b_{\rm{max}}}{(GM_{\rm{BH}})/v_{\rm{BH}}^{2}}$ (2)

with $b_{\rm{max}}=20kpc$, and $\mathcal{F}$ is

$\mathcal{F}={\rm{erf}}(x)-\frac{2x}{\sqrt{x}}e^{-x^{2}},x=\frac{v_{\rm{BH}}}{\sigma_{v}}$
(3)

from integrating the Maxwellian distribution. This dynamical friction
implementation produces physically realistic motion for black holes due to
small scale interactions with the nearby matter, and stabilizes the black hole
once it reaches the galactic center, which provide more realistic information
for the black holes leading up to mergers. For a more detailed discussion of
this implementation and the black hole orbital information it produces, see Ni
et al. (2022); Chen et al. (2022a).

##  3 Black hole / merger populations

Figure 1: The black hole mass function for Astrid (red) and TNG300 (blue), at
z=2-7. For each simulation, the black hole seed model is clearly seen at the
lowest masses, but above the seed mass the two simulations broadly agree for
$z>4$, though TNG300 produces more high-mass black holes at late times.

Before investigating mergers in the Astrid simulation, we first consider the
overall populations of black holes over cosmic time. In Figure 1 we plot the
black hole mass function (BHMF) at $z=$2, 3, 4, 5, 6, and 7 (solid, dashed,
dotted, and dot-dashed, respectively) from the Astrid simulation (red), with a
comparison to the TNG300 mass function (blue) at the same redshifts. We find
the high end of the mass function ($M_{\rm{BH}}>3\times 10^{6}M_{\odot}$)
tends to follow an approximate power law with a slope of $\sim-1.05$ (for
$M_{\rm{BH}}>10^{6}M_{\odot}$). The slope is slightly steeper at earlier
times, but the primary evolution is the increase in normalization (increasing
by more than 2 dex from z=7 to z=2; see Table 1 for best fitting power-law
parameters at each redshift). The TNG300 BHMF is broadly consistent with these
Astrid results, except that TNG300 produces more high-mass black holes at
later times ($\sim 3\times$ more $\sim 10^{8}\ h^{-1}\ M_{\odot}$ black holes
at z=4). As such, we can see that massive black holes are able to grow more
efficiently in TNG than Astrid at late times (except for the most-massive
end), which we can expect to have some impact on the merging populations as
well. However, we note that this will only affect the largest black holes at
the latest times, and thus will not influence the majority of mergers when
comparing the two simulations.

At lower masses, we see that the two simulations diverge significantly, as a
result of the seeding model. In the TNG300 simulation, we see a large spike at
the lowest mass BHs ($\sim 10^{6}M_{\odot}$). This peak corresponds to the
black hole seed mass used in the TNG simulations, and is a result of recently-
seeded black holes tending to have very low accretion rates due to stellar
feedback, especially at early times (Weinberger et al., 2017, 2018); hence a
large number of black holes have not grown much beyond their seed masses. In
contrast, the Astrid BHMF gets slightly steeper at lower masses, but does not
have any qualitative shift until $M_{\rm{BH}}\sim 10^{5.5}M_{\odot}$, below
which we find a shallower power law. We recall that the seeding model used in
Astrid initializes black holes with a starting mass selected from a power-law
distribution with slope of -1 (see Section 2), which produces the behaviour we
see here. We note that the best fitting slope for this plateau is slightly
shallower than the slope for the seed mass selection ($\sim-0.9$ rather than
the $-1$ used when seeding), and gets gradually shallower with time (from
$\sim-0.96$ at z=7 to $\sim-0.5$ at z=2; see also Table 1). This is due to
lower-mass seeds gradually growing into the higher mass range for seeding, and
hence over time the number of black holes in the high-mass end of the seed
range increases. As such the plateau we see in the BHMF is a result of the
seed model used combined with a small amount of growth, while for higher-mass
black holes (e.g. $>10^{6}M_{\odot}$) we find a well-behaved BHMF which is
consistent with the TNG BHMF.

Figure 2: Merger rate signal as a function of redshift in Astrid (red), TNG300
(blue), TNG100 (yellow), and TNG50 (purple). Since Astrid includes much lower
mass black holes than the TNG simulations, the dashed red line shows the
Astrid signal rate when limited to only mergers massive enough to be resolved
in TNG300. We see that all five simulations have qualitatively similar
behaviour, though with different normalization: Astrid has significantly more
mergers (due to the lower-mass black holes). When considering only mergers
between blackholes above $2\times M_{\rm{TNG,seed}}$ (dotted lines), however,
we find a lower merger rate in Astrid.  Figure 3: Distribution of primary
merger mass ($M_{1}$) and redshift for Astrid (left) and TNG300 (right;
restricted to $z>2$ to match Astrid). The different seed models allows Astrid
to probe to lower masses and higher redshifts, but produce comparable results
above $M_{\rm{seed,TNG}}$.

Overall, we find that Astrid is similar to TNG300 (and comparable
simulations), though with two significant differences. First, Astrid spans a
larger range of black hole masses, extending to much smaller black holes
resulting from the new seed model. Additionally, Astrid has an improved black
hole treatment, including dynamical friction, which more accurately models
black hole motion and mergers. As such, we expect the merger rates to be more
strongly affected, and a potential impact on high-mass black holes (after
longer time to grow), hence the discrepancy between Astrid and TNG300 in
Figure 1 increases for late times and high masses.

Next we consider the mergers produced by this black hole population. In Figure
2 we plot the rate at which GW signals will reach the Earth from SMBH mergers,
obtained by integrating the number of mergers in the simulation over
redshifts, incorporating the cosmic volume at the given redshift:

$\frac{{\rm d}N}{{\rm d}z\,{\rm
d}t}=\frac{1}{z_{2}-z_{1}}\int_{z_{1}}^{z_{2}}\frac{{\rm d}^{2}n(z)}{{\rm
d}z\,{\rm d}V_{c}}\frac{{\rm d}z}{{\rm d}t}\frac{{\rm d}V_{c}}{{\rm
d}z}\frac{{\rm d}z}{1+z}\,$ (4)

We see that the TNG simulations (TNG300 - blue; TNG100 - yellow; TNG50 -
purple) have broadly comparable behaviour: very rare mergers at high z,
increasing with time to a peak at z$\sim$2, followed by a decrease in expected
merger rates. The high-redshift behaviour in Astrid (red) is broadly similar,
in that it starts with rare mergers at high z, and increases with time.
However, we find an overall rate significantly higher in Astrid than any of
the TNG simulations. Rather than a fundamental difference in mergers, however,
this increased rate in Astrid is a result of the seed model initializing lower
mass black holes in lower mass galaxies when compared to TNG. As shown in
Figure 1, at high masses the simulations are consistent, but Astrid includes
many low-mass BHs below TNG’s seed mass, and hence will include a significant
number of mergers which go unresolved in TNG.

| High-mass | Low-mass
---|---|---
z | $a$ | $b$ | $c$ | $d$
2 | -2.7 | -1.0 | -1.6 | -0.49
3 | -3.3 | -1.0 | -2.0 | -0.60
4 | -3.9 | -1.1 | -2.5 | -0.79
5 | -4.6 | -1.0 | -2.9 | -0.87
6 | -5.4 | -1.1 | -3.3 | -0.93
7 | -6.3 | -1.1 | -3.6 | -0.96
Table 1: Best fitting parameters for black hole mass function (Figure 1),
fitting $M_{\rm{BH}}>10^{6}M_{\odot}$ to $10^{a}(M/10^{7}M_{\odot})^{b}$, and
$M_{\rm{BH}}<10^{5.5}M_{\odot}$ (seed-mass range) to
$10^{c}(M/10^{7}M_{\odot})^{d}$.

To test this, the dashed lines in Figure 2 show the rates from the Astrid and
TNG300 simulations after imposing a cut of $2\times M_{\rm{seed,TNG}}$ (i.e.
only mergers with $M_{1}>M_{\rm{seed,TNG}}$ and $M_{2}>M_{\rm{seed,TNG}}$,
where $M_{1}$ ($M_{2}$) is the larger (smaller) mass involved in a merger, and
$M_{\rm{seed,TNG}}$ is the black hole seed mass for the TNG simulations). A
cut of $M>2\times M_{\rm{seed,TNG}}$ removes black holes which are not yet
seeded by TNG, and also ignores the strong peak in the TNG BH population
caused by the seed criteria, and thus we are comparing equivalent black hole
populations. Here we see that the majority of mergers involve low mass BHs
(where at least one BH has $M_{\rm{BH}}<2\times M_{\rm{seed,TNG}}$, hence
dashed line well below the solid lines), and when limited only to the
equivalent merger populations, Astrid actually has a lower merger rate than
the TNG simulations by a factor of $\sim 3$.

There are several major differences between the simulations which have the
potential to influence the merger rates. Once factor is that Astrid seeds
black holes into smaller halos, and the dependence of galaxy merger rates on
galaxy mass will influence the rate at which BHs seeded in those galaxies end
up merging. However, this will be limited primarily to low mass galaxies/BHs,
and so would not explain the lower rate in Astrid among high-mass mergers.
Another factor is the faster growth in TNG, which produces a larger population
of high-mass black holes when compared to Astrid (see Figure 1); at $z=4$ TNG
has $\sim 50\%$ more black holes above $2\times M_{\rm{seed,TNG}}$. Hence we
should expect a higher merger rate in TNG than Astrid, based solely on the
population of black holes available to merge. Finally, there is the dynamical
friction model: Astrid models black hole motion using a dynamical friction
prescription (Chen et al., 2022b) rather than a recentering scheme. As such,
when a pair of galaxies merge together, satellite BHs (those found in the
smaller of the two merging galaxies) can take longer to reach the galaxy
center where they are then able to merge with the central BH, and the full
dynamical friction model allows for flyby interactions rather than assuming an
immediate merger when two black holes are sufficiently close together. Thus we
expect black holes in simulations which incorporate dynamical friction to
generally merge more slowly than those in simulations which use a recentering
scheme. They should therefore have a lower overall merger rate (when
controlling for resolved masses), precisely as we see when comparing Astrid to
TNG300 (dotted lines in Figure 2).

To more directly consider merging masses, we plot the mass-redshift
distribution of mergers in Figure 3 for both Astrid (left) and TNG300 (right).
Consistent with Figures 1 & 2, we see that the lower seed mass in Astrid
provides $M_{1}$ more than an order of magnitude below what the TNG
simulations resolve. Similarly, the lower halo mass threshold for seeding
means that Astrid models black hole mergers out to earlier cosmic times, as a
result of seeding black holes into smaller galaxies. Above the seed mass
scales, however, we see that both Astrid and TNG300 produce remarkably similar
distributions, with $M_{1}>M_{\rm{seed,TNG}}$ mergers first occurring at
$z\sim 7$, and reaching $M_{1}\sim 10^{10}M_{\odot}$ at z=2 (though Astrid’s
larger volume means there are a few unusually massive mergers at slightly
earlier times). This is consistent with Figure 1, in that black holes well
above the seed mass tend to grow at comparable rates in both Astrid and the
TNG simulations at high-z, and thus the mass scale involved in the mergers
tends to be similar. Combined with Figure 1, we see that the smaller seeds in
Astrid (and the power-law seed mass distribution) are capable of growing to
the masses necessary to match high-redshift black hole observations and
produce black hole populations fully consistent with current constraints. A
complete analysis of typical black hole growth behaviours is beyond the scope
of this paper, but will be discussed in an upcoming work.

Figure 4: BHMF (solid) for Astrid (red) and TNG300 (blue) at z=2 (top) to z=5
(bottom), compared to the merger mass function based on the previous 75 Myr
(dashed; see text for details). In both simulations the merger mass functions
are similar, though the Astrid merger mass function is smaller relative to the
overall BHMF, suggesting a lower merger rate.

To compare merger rates to black hole populations, in Figure 4 we plot the
black hole mass function (solid lines), and compare to the merger mass
function (dashed line) at $z=2,3,4,5$. We define the merger mass function to
be the mass function of $M_{1}$ (the mass of the more massive BH involved in a
merger) for all mergers which take place in the previous 75 Myr. Here we see
that the merger mass function in each simulation is comparable to (though
slightly lower than) the corresponding BHMF. In the case of TNG300, we see
that above the seed mass both the BHMF and the merger mass function follow a
rough power law, and both functions also show a peak at the seed mass (though
that spike is smaller in the merger mass function). Similarly, in Astrid we
see a rough power law above $\sim 10^{6}M_{\odot}$, and a plateau below $\sim
10^{5.5.}M_{\odot}$ which corresponds to the power-law seeding model. Similar
to TNG, we see that at the seed-mass scale the merger mass function is
significantly below the BHMF: although there is a large population of recently
seeded black holes, they are less likely to undergo a merger (as they
necessarily need time for their host halos to merge, during which they are
able to grow past their seed mass).

Figure 5: The typical merger rate for a BH with a given mass in the Astrid
(red) and TNG300 (blue) simulations, as well as the merger rates limited to
$M_{1,2}>2*M_{\rm{seed,TNG}}$ (dotted lines). Both simulations have comparable
total merger rates, but the majority of mergers in Astrid involve a low-mass
secondary BH; when restricted to the same secondary mass range, Astrid has a
merger rate $\sim 1$ dex lower than TNG300. [NOTE: This is not the overall
rate of mergers in the simulation/universe; rather it is the rate at which an
individual BH of a given mass would be expected to undergo mergers; i.e. lower
panel would suggest a z=5 BH with M$\sim 10^{6.5}M_{\odot}$ would be expected
to undergo about 1 merger per Gyr. That’s true for both TNG and Astrid, but in
Astrid that includes mergers with lower-mass BHs (below the seed mass of TNG);
the dashed line only includes mergers where both BHs are above the seed mass
of TNG, so it’s closer to a 1-to-1 comparison.] Figure 6: Frequency-strain
signals for GWs emitted by BH mergers over cosmic time (binned data) against
the LISA sensitivity curve (dashed black line). Each bin is colour-coded by
the mean redshift of the source merger which emitted the given GWs. (a): All
signals from the Astrid simulation (limited to $z>2$). (b): All signals from
the TNG300 simulation, which is complete to z=0 (hence note the different
colourscale range), but limited to higher masses. (c): Signals from the Astrid
simulation, but limited to the same mass scales as TNG300. (d): Signals from
the TNG300 simulation, but limited to the same redshift range ($z>2$) as the
current state of the Astrid simulation.

Although the ratio between BHMF and merger mass function is qualitatively
similar between Astrid and TNG, suggesting a comparable merger rate, when
selecting BHs by mass. In Figure 5, we have divided the merger mass function
by the BHMF to obtain a characteristic merger rate as a function of
$M_{\rm{BH}}$ (i.e. the typical number of mergers that a BH with a given mass
would undergo in 1 Gyr). As discussed in reference to Figure 4, we expect the
merger rate for BHs near the seed mass to generally be much lower than the
merger rate for higher mass black holes: we see this explicitly in Figure 5,
as both Astrid (red) and TNG300 (blue) show a significant dropoff near their
respective seed masses. Above the seed masses, however, both simulations show
an approximate power law relating merger rate to $M_{\rm{BH}}$, such that
high-mass black holes have a slightly higher merger rate than lower-mass black
holes. Furthermore, we see that both Astrid and TNG300 have comparable merger
rates; however, we note that this is somewhat coincidental, based on near
seed-mass mergers. As seen in Figures 1-4, Astrid contains a large number of
low-mass black holes which are unresolved in TNG, but which contribute to the
merger rates show in Figure 5. We show this explicitly with the dashed line,
where we only consider mergers in which $M_{2}>2\times M_{\rm{seed,TNG}}$, in
which case we have roughly an order of magnitude fewer mergers. Nonetheless,
after removing the low-mass (and thus less well-resolved) mergers, we find
fairly comparable merger rates except at the highest masses, where Astrid
predicts a slightly longer time between mergers.

##  4 Gravitational Wave Signals

In addition to the underlying black hole merger populations, we consider the
gravitational waves emitted by calculating both the frequency and strain of
the GW for each merger. We use the characteristic strain, $h_{s}$, to model
the binary signal which accounts for the time the binary spends in each
frequency bin (Finn & Thorne, 2000). The characteristic strain is given by
(e.g. Moore et al., 2015):

$h_{s}(f)=4f^{2}|\tilde{h}(f)|^{2}$ (5)

where $\tilde{h}(f)$ represents the Fourier transform of a time domain signal.
To generate the waveforms, we use the phenomenological waveform PhenomD (Husa
et al., 2016; Khan et al., 2016) implemented within the gwsnrcalc Python
package (Katz & Larson, 2019). The input parameters are the binary masses,
merging redshift, and the dimensionless spins of the binary. For the SMBH
masses, we do not account for mass growth after the numerical merger. However,
we note that the SMBH can potentially gain a significant fraction of its mass
during the $>1\,{\rm Gyr}$ of time in the dynamical friction (e.g. Banks et
al., 2022) or loss-cone scattering phase. The dimensionless spin $a$
characterizes the alignment of the spin angular momentum with the orbital
angular momentum, and the value of $a$ ranges from $-1$ to $1$. However, we do
not have any information on the spin of the SMBHs in our simulation.
Therefore, following the argument in Katz et al. (2020), we assume a constant
dimensionless spin of $a_{1}=a_{2}=0.8$ for all binaries (e.g. Miller, 2007;
Reynolds, 2013).

In Figure 6 we plot the range of frequencies and strains for GW signals
emitted by mergers in Astrid (panel a), with each frequency-strain bin colour-
coded by the mean redshift of the emitting mergers. We see that the majority
of GW signals come from the lowest-redshift mergers (though note that the
lowest redshift here means $z\sim 2$, the latest time probed in this
analysis), since the merger rate increases with time (at least at high-z; see
Figure 2). We note that high-z does dominate the lowest-strain GW signals,
which is not only a direct result the merger being more distant (and hence a
weaker signal), but also because high-z mergers tend to involve swallowing a
lower-$M_{2}$ BH (which necessarily produces a smaller strain compared to
swallowing a higher-mass secondary BH). We also see an upper bound on the GW
signals, however we note that is a result of the limited redshift window
reached so far by Astrid (as the simulation continues to lower-z, the upper
right area will continue to be filled in, which we see in panels $b$ and $d$
for TNG300 mergers).

In Figure 6b, we show the frequency-strain signals for the TNG300 mergers,
again colour-coded by the merger redshift (though note the range for the
colour bar is different, since TNG is complete to z=0). As in Figure 6a, we
see that most signals come from the lowest redshifts (although here that means
near $z=0$), and again high-z signals dominate the lowest-strains. However,
the much larger redshift range makes comparing to these panels problematic;
instead, in Figure 6d we plot the GW signals from TNG300, but restricted to
the same redshift range as Astrid ($z>2$). Here we see a result which is more
similar to the Astrid data (Figure 6a), but which spans a much more limited
range of frequency and strain. The z$\sim$2 limit imposes a strict upper limit
on the strain, and we see that both Astrid (Figure 6a) and TNG300 (Figure 6d)
have similar distributions when limited to the same redshift range, with the
exception that Astrid extends higher frequencies and lower strain, which we
expect since Astrid includes black holes below the resolved mass in TNG300.
Overall, we see that the GW signals tend to be dominated by the lowest
redshift mergers, with an exception for the highest frequency/lowest strain
signals, which tend to be at higher redshifts.

Unlike Figure 6a, which spans the majority of the LISA frequency range, the
frequency-strain distribution for TNG300 (Figure 6b & d) is limited only to
LISAs low-frequency regime. This is a result of Astrid resolving much lower
black hole masses, whose mergers produce higher frequency signals. We see this
explicitly in Figure 6c, which shows the GW signals in Astrid, but limited to
the same mass range as TNG300 (i.e. $M_{2}>M_{\rm{seed,TNG}}$), such that
Figure 6c & d show the results from Astrid and TNG300 for the same mass and
redshift ranges. Comparing these panels, we see that Astrid has fewer mergers
(at fixed mass/redshift ranges) than TNG300, consistent with Figures 2 & 5
which also show fewer mergers. Since the mass scale is well above the seed
mass, this is not a result of the seeding prescriptions. Rather it is a result
of the dynamical friction modeling which delays black hole mergers to later
times relative to a simpler repositioning scheme. Overall, we see that the
largest difference between the two simulations is that Astrid produces more
high-frequency and low-strain signals, which correspond to low-mass mergers.

Figure 7: Distribution of signal-to-noise ratios (SNR) for $z>2$ mergers from
Astrid and TNG300, normalized by simulation volume (solid lines). Both
simulations show qualitatively similar behaviour, with the distribution
peaking at high SNR. Because the simulations use different seed masses, we
also compare the SNR distribution for mergers in which both merging black
holes are above $2\times$ the TNG seed mass (dashed lines), as well as the
distribution of Astrid signals for mergers in which both merging black holes
are above $10^{5.5}\ h^{-1}\ M_{\odot}$ (the high-end of Astrid’s seed mass
range).  Figure 8: Distribution of merging black hole masses for Astrid (left)
and TNG300 (right), color coded by the mean log(SNR) for the mergers. We find
that the strongest SNRs in Astrid are found for $M_{1},M_{2}\sim
10^{6}M_{\odot}$; this is consistent with TNG300, although the higher seed
mass in the TNG simulations means that the peak signals roughly correspond to
the seed masses of the simulation. In both simulations, we also note that the
SNR is more strongly correlated to $M_{1}$ than $M_{2}$.

###  4.1 SNR

Having calculated the GW frequency and strain emitted by merging black holes,
we also consider the strength of the GW signal received by LISA. We estimate
the Signal-to-Noise (SNR) by integrating the ratio of the signal to the noise
in the frequency domain. The sky, orientation, and polarization averaged SNR
are given by :

$\langle{\rm SNR}\rangle^{2}=\frac{16}{5}\int_{f_{\rm start}}^{f_{\rm
end}}\frac{h_{s}^{2}}{h_{N}^{2}}f^{-1}df,$ (6)

where $f_{\rm start}=f(t_{\rm start})$ and $f_{\rm end}=f(t_{\rm end})$, with
$t_{\rm start}$ and $t_{\rm end}$ representing the starting and ending time
when the signal is observed. Note that here we do not account for the
eccentricity of the binaries, and assume circular orbits at the time of
merger.

Figure 9: Distribution of SNR vs. mass (left: $M_{1}$; right: $M_{2}$) for
$z>3$ mergers in Astrid (top) and TNG300 (bottom), color-coded by density of
mergers in the simulation volume. Consistent with Figure 8, we see that SNR is
much more strongly correlated wtih $M_{1}$ than $M_{2}$, and we again find
that Astrid and TNG300 are broadly consistent above the TNG seed mass.

For the current configuration, we assume that the LISA observation lasts for 4
years. We further assume a most optimistic SNR for all mergers by taking
$t_{\rm end}=t_{\rm peak}$ and $t_{\rm start}=t_{\rm peak}-4{\rm yrs}$. Under
this assumption, we are always integrating the part of the waveform where the
strain is maximized. However, as was discussed in Salcido et al. (2016) and
Katz et al. (2020), the actual SNR may be smaller if there is an offset
between the LISA observation window and the merger time of the binary.

In Figure 7, we plot the distribution of SNRs from Astrid (solid red) and
TNG300 (solid blue) for $z>2$ mergers, normalized by the simulation volume. In
both simulations we find that the SNR peaks at very high values
($10^{2}-10^{3}$), with Astrid producing significantly more high-SNR mergers,
while TNG300 has a flatter tail extending toward low-SNR. Since the majority
of mergers tend to occur between low-$M_{\rm{BH}}$ black holes (e.g., see
Figures 3-4), these differences are expected due to the different seed
criteria between the two simulations. For a more direct comparison, the dashed
lines show the SNR distribution limited to mergers in which both merging black
holes have $M_{\rm{BH}}>2\times M_{\rm{TNG,seed}}$ (i.e. only including BHs
more than double the TNG seed mass, to avoid issues related to black hole
seeding). Here we see that the two simulations have a consistent peak at SNR
just below $10^{3}$, though we find that TNG300 has a nearly flat distribution
for SNR$>$1, whereas Astrid predicts a moderate slope, producing fewer low-SNR
mergers. This appears to be due to TNG300 producing more high-mass black holes
at $z\leq 4$ (see Figure 4) as a result of more efficient black hole growth. A
detailed comparison of black hole growth efficiency between these simulations
is beyond the scope of this paper, and is left for a future work. However, we
note that the difference only affects the low-SNR end, and will thus have a
comparatively minor impact on the LISA detection rate.

On the other hand, the high-SNR end is clearly significantly impacted by the
seeding criteria. Although Astrid and TNG agree when limited to merging masses
about $2\times M_{\rm{TNG,seed}}$, the majority of mergers are removed when
applying such a high mass cut. One of the major advantages of the Astrid
simulation is that the lower seed masses allow us to effectively probe lower
mass mergers. We show this with the dotted line in Figure 7, where we consider
any mergers between BHs above the Astrid seed mass distribution (i.e.
$M_{\rm{BH}}>10^{5.5}\ h^{-1}\ M_{\odot}$, hence above the seed-mass dominated
regime, as seen in Figure 1), which provides us with significantly more
mergers. We note that including these lower-mass black holes produces a peak
at the high-SNR end which is missed when imposing a higher mass cut (or higher
seed masses). Furthermore, we find roughly triple the number of SNR$<10^{2.5}$
for $M_{2}>10^{5.5}\ h^{-1}\ M_{\odot}$ than for $M_{2}>2\times
M_{\rm{seed,TNG}}$, emphasizing the importance of low-mass black holes.
Furthermore, we note that the slope of the SNR$<10^{2.5}$ distribution remains
roughly the same when using the lower mass cut. Overall, this suggests that
the highest-SNR mergers are primarily between two low-mass BHs which are
completely missed with a higher mass cut (hence the strong peak which the
dashed line completely misses), while the lower-SNR mergers frequently involve
at least one higher-mass black hole, so by including a smaller mass cut we
increase the number of mergers at all SNR-ranges but without affecting the
slope.

We investigate this in more detail in Figure 8, which shows the mass
distribution of merging black holes ($M_{1}$ vs. $M_{2}$), color coded by
$<\rm{log}(SNR)>$ (i.e. the mean logarithmic SNR for mergers in the given mass
bin), for $z>3$ mergers in Astrid (left) and TNG300 (right). In the mass
scales resolved in both simulations we find comparable SNR results (though we
again see more high-mass mergers in TNG as a result of the more efficient
growth). As seen in Figure 7, the larger mass range produced by Astrid’s
smaller seed mass results in significantly more mergers, especially for high
SNRs. In particular, the Astrid simulation shows the SNR peaks at
$M_{1},M_{2}\sim 10^{6}M_{\odot}$, with the lower masses having weaker (though
still strong) signals (they are below LISA’s peak sensitivity, but still well
within the detectable range).

Figure 8 also shows that SNR depends primarily on the larger mass ($M_{1}$),
while the secondary black hole ($M_{2}$) has a relatively weak impact on SNR.
We show this explicitly in Figure 9, which plots the correlation between SNR
and the merging black hole masses ($M_{1}$ \- left; $M_{2}$ \- right) for
Astrid (top) and TNG300 (bottom), color coded by the number density of
mergers. The majority of mergers occur between low mass black holes (as seen,
e.g., in Figure 4), and we again see that the highest SNRs occur when both
$M_{1}$ and $M_{2}$ are $\sim 10^{6}M_{\odot}$. We also see that SNR
correlates quite strongly with $M_{1}$, spanning $\sim$1 dex at $M_{1}\sim
10^{6}M_{\odot}$ up to $\sim$2 dex at higher masses. In contrast, a low
$M_{2}$ spans a much higher range of SNRs, corresponding to the wide range of
possible $M_{1}$ (i.e. mass of the swallowing black hole), though most
low-$M_{2}$ mergers still have high-SNR (since the majority of mergers will
still have a low $M_{1}$). The relatively weak dependence on $M_{2}$ means
that the growth of the secondary BH does not play a significant role in the
expected SNR for a given merger. However, it is nonetheless very important to
include lower mass black holes to probe the full range of mergers which we
expect LISA to be able to detect: mergers between $\sim 10^{6}M_{\odot}$ black
holes are both extremely common and produce the strongest gravitational wave
signals from LISA, so limiting simulations to higher masses will necessarily
miss the majority of mergers and their associated GW signals. On the other
hand, we note that the LISA SNR peaks at both $M_{1}\sim 10^{6}M_{\odot}$ and
$M_{2}\sim 10^{6}M_{\odot}$. So although incorporating smaller seed masses in
simulations will provide more accurate estimates for the overall merger rates
(which are dominated by low-mass mergers), below $\sim 10^{5}M_{\odot}$ will
primarily consist of lower-SNR signals.

##  5 Conclusions

In this work, we have investigated the black hole population in the Astrid
simulation, which seeds black holes at masses as low as $10^{4.5}\ h^{-1}\
M_{\odot}$, well below what is frequently used in comparable cosmological
simulations. We have particularly focused on the mergers between black holes,
looking at the overall merger rates, the merging masses, and the expected
gravitational wave signals they produce, noting that Astrid directly
incorporates a dynamical friction model which produces more realistic merger
behaviour. Our main results are as follows:

  * •

The Astrid simulation produces a comparable population of high-redshift black
holes when compared to prior simulations (esp. TNG300), but with an alternate
seed model which extends to much lower mass black holes. Above the masses
resolved by both simulations, we see that Astrid and TNG have comparable mass
functions (Figure 1), and also typical merging masses as a function of
redshift (Figure 3).

  * •

The overall merger rate is much higher in Astrid than in earlier TNG
simulations, primarily due to mergers involving low-mass black holes which are
not resolved in simulations with higher seed masses. When considering the same
mass scales, Astrid has fewer mergers, likely a result of the added infall
time resulting from the dynamical friction model incorporated in Astrid.

  * •

The merger rate for a given black hole is mass-dependent, with massive black
holes undergoing mergers more frequently than low-mass black holes. A typical
black hole can be expected to undergo a merger every $\sim$1-10 Gyr.

  * •

Including low-mass black holes (i.e. lower mass seeding prescriptions) is
crucial for modeling LISA detections. High-mass seed models will only probe
the low-frequency, high-strain regime within the LISA sensitivity band, while
lower seed masses can extend across the full range of potential LISA
detections.

  * •

In addition to decreasing the overall merger rate, including dynamical
friction can be expected to preferentially affect low-frequency, low-strain GW
signals, which are primarily generated by low-$M_{2}$ mergers. This further
emphasizes the importance of including accurate models for both seeding and
infall dynamics, as both have the potential to not only affect the expected
signal rate, but also the frequency-strain distribution of GW signals that
LISA will be expected to detect.

  * •

The SNR distribution follows a rough power law until the highest signals (SNR
$>10^{2}$), at which point there is a peak caused by $M_{\rm{BH}}\sim
10^{6}M_{\odot}$ mergers. This peak is poorly resolved in simulations with
high seed masses, but is 0.5-1.5 dex above the Astrid seed masses, providing a
well resolved sample. Additionally, SNR is strongly correlated with the more
massive black hole ($M_{1}$), and weakly correlated to the less massive black
hole ($M_{2}$; though note that $M_{1}$ and $M_{2}$ are correlated to each
other).

  * •

The LISA SNR has a weak dependence on redshift, at least in the early universe
($z>2$). There is evolution in the merging populations of black holes, with
later times including higher-mass (and correspondingly smaller SNR) mergers.
However, low mass mergers have stronger signals, are more common, and have
relatively weak dependence on redshift (at least for $z>3$, probed here).

In summary, we have shown that using a wide range of black hole seed masses
extending down to $10^{4.5}\ h^{-1}\ M_{\odot}$ produces a high-mass
population of black holes comparable to simulations which use higher seed
masses, while simultaneously allowing us to investigate smaller black holes,
mergers, and the associated gravitational waves they emit. In particular, we
are able to provide a detailed investigation into mergers spanning much of the
LISA sensitivity range, provide estimates for the expected GW signal
detections, and the correlation between these detections and the underlying
merger masses and redshifts, in a simulation which directly models dynamical
friction to produce more accurate merger behaviour.

## Acknowledgements

Astrid was run on the Frontera facility at the Texas Advanced Computing
Center. TDM acknowledges funding from the NSF AI Institute: Physics of the
Future, NSF PHY-2020295, NASA ATP NNX17AK56G, and NASA ATP 80NSSC18K101. TDM
acknowledges additional support from NSF ACI-1614853, NSF AST-1616168, NASA
ATP 19-ATP19-0084, and NASA ATP 80NSSC20K0519. SPB was supported by NASA ATP
80NSSC22K1897.

## Data Availability

The code used for the simulation is available at https://github.com/MP-
Gadget/MP-Gadget. Halo catalogs and BH data are available upon reasonable
request to the authors.

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##  Appendix A Frequency Strain counts

Figure 10: Same as Figure 6, but colour coded by the number density of
mergers, rather than the mean redshift. We see that the merger frequency tends
to peak toward higher-frequencies (corresponding to lower merger masses), and
thus higher LISA SNR. This is strongest when considering lower mass mergers
(e.g. when considering mergers down to the Astrid seed mass scale, in the top
left panel), or when limited to high redshift (when black holes tend to be
smaller).

Similar to Figure 6, in Figure 10 we show the frequency-strain distribution
for both the Astrid and TNG300 simulations, but here color-coded by the number
density of mergers. Here we see that the peak in the frequency-strain
distribution occurs at the highest frequencies, and lowest strain,
corresponding to low-mass mergers taking place at low redshift, consistent
with the merger mass functions in Figure 4. Here we again see the importance
of including lower-mass black hole seeds, as a higher seed mass will cut out
the majority of mergers, which also correspond to the peak sensitivity range
for LISA (compare Figure 10 panel a to panel c). By comparing Astrid to TNG,
we also note that the higher seed mass results in a higher number of low-
frequency GWs, especially at the lowest redshifts.