Probing Supermassive Black Hole Seed Scenarios with Gravitational-wave Measurements

The process whereby the supermassive black holes (SMBHs) populating the centers of galaxies have been assembled remains to be established, with the relative importance of seeds provided by collapsed Population III stars, black holes formed in nuclear star clusters via repeated mergers, or direct collapses of protogalactic disks yet to be determined. In this paper we study the prospects for casting light on this issue by future measurements of gravitational waves emitted during the inspirals and mergers of pairs of intermediate-mass black holes (IMBHs), discussing in particular the roles of prospective measurements by LISA and the proposed atom interferometers AION and AEDGE. We find that the expected number of detectable IMBH binaries is O(100) for LISA and AEDGE and O(10) for AION in low-mass seeds scenarios and goes down to O(10) for LISA and below one for AEDGE and AION in high-mass seed scenarios. This allows all of these observatories to probe the parameters of the seed model, in particular, if at least a fraction of the SMBHs arises from a low-mass seed population. We also show that the measurement accuracy of the binary parameters is, in general, best for AEDGE, which sees very precisely the merger of the binary.


INTRODUCTION
Most galaxies contain supermassive black holes (SMBHs) heavier than 10 6 M ⊙ (Kormendy & Ho 2013), and the existence of black holes (BHs) with masses between a few and ∼ 80M ⊙ has been established by observations of X-ray binaries (Remillard & McClintock 2006) and by the measurements of gravitational waves (GWs) with frequencies ∼ 100 Hz emitted during their mergers (Abbott et al. 2019(Abbott et al. , 2021a,b),b).Various other observations point to the existence of intermediate-mass black john.ellis@cern.chmalcolm.fairbairn@kcl.ac.uk juan.urrutia@kbfi.eeville.vaskonen@pd.infn.itholes (IMBHs) with masses in the range 10 4 − 10 6 M ⊙ , but their mass function and redshift distribution is known only very poorly (Greene et al. 2020).This lack of information about IMBHs impedes our understanding of how SMBHs have been assembled (Reines 2022).
The main possibilities for seeding SMBH assembly include collapsed Population-III stars (Madau & Rees 2001;Bromm & Loeb 2003), BHs formed in nuclear star clusters via repeated mergers (Portegies Zwart et al. 2004;Atakan Gurkan et al. 2004;Natarajan 2021) or direct collapses of protogalactic disks in which fragmentation is suppressed (Sesana et al. 2004;Begelman et al. 2006;Volonteri et al. 2008;Mayer et al. 2007;Tanaka & Li 2014;Inayoshi et al. 2015;Izquierdo-Villalba et al. 2023).All of these mechanisms are capable of reproducing the properties of the observed SMBH population for suitable values of assembly parameters such as accretion rates, but can differ significantly in their pre-dictions for the spectrum of IMBH mergers at different redshifts, see Volonteri et al. (2021) for an overview.Signatures of these mechanisms may include either light seeds, with masses ∼ 10 2 − 10 3 M ⊙ or heavy seeds, with masses ∼ 10 4 − 10 5 M ⊙ , at z ≲ 10.These are currently unconstrained by data, but can in principle be probed by future GW and other measurements (Sesana et al. 2011;Hartwig et al. 2016;Krolik et al. 2019;Mangiagli et al. 2020;Volonteri et al. 2020;Haidar et al. 2022;Dong-Páez et al. 2023).
The purpose of this paper is to investigate what progress can be made in distinguishing between the SMBH assembly scenarios with planned future GW experiments at frequencies larger than 0.1 mHz in light of recent data on nHz GWs from pulsar timing array (PTA) experiments, highlighting the potential capabilities of atom interferometers.Whereas PTAs are sensitive to the early inspiralling phase of the SMBHs heavier than 10 9 M ⊙ , the future GW experiments will probe directly the infalls and mergers of binaries lighter than 10 7 M ⊙ .
NANOGrav (Agazie et al. 2023a) and other PTA experiments (Antoniadis et al. 2023a;Zic et al. 2023;Xu et al. 2023) have recently reported the observation of a stochastic background of GWs at frequencies in the nHz range, for which the most conservative astrophysical interpretation is that binary systems of SMBHs are emitting them with masses ∼ 10 9 M ⊙ (Agazie et al. 2023b,c;Antoniadis et al. 2023b;Ellis et al. 2023b).Naive extrapolation of binary merger models to lower BH masses suggests that GWs from IMBH binaries may be observable at higher frequencies between 0.1 mHz and 1 Hz, for example by the LISA space-borne laser interferometer experiment or atom interferometer experiments (Ellis et al. 2023a).In principle, there are two regimes where the formation channels of the SMBHs may be distinguished.Either at z ≳ 7 when the seeds are assembling and scaling relations are not that strong (Treister et al. 2013;Ricarte & Natarajan 2018a), or by observing the low mass occupation fraction of dwarf galaxies and dark matter halos at more recent times (Ricarte & Natarajan 2018b;Volonteri et al. 2021).The extrapolation to higher frequencies of a model that can fit the NANOGrav background (Ellis et al. 2023b) predicts that the majority of detectable binaries will be at z < 7, so the focus of this study is to constrain the latter.Our work complements analogous studies that have been performed with electromagnetic observations of active galactic nuclei (AGNs) (Kelly & Shen 2013;Miller et al. 2015;Gallo & Sesana 2019;Chadayammuri et al. 2023).
To extend these observations to the GW spectrum and establish a multi-messenger signal, we estimate how well the space-borne laser interferometer experiment LISA (Amaro-Seoane et al. 2017) and the atom interferometers AION-1km (Badurina et al. 2020(Badurina et al. , 2021) ) and AEDGE (El-Neaj et al. 2020;Badurina et al. 2021) could recover the occupation fraction and the weight of the different channels contributing to the formation of the SMBHs.1 Our approach is to generate populations of the IMBH binaries, estimate how well the binary parameters could be recovered, and then construct posteriors for the parameters of the merger rate.To extend the deduced merger rate from the NANOGrav observations, we assume that the scaling relation extends to lower masses in a light-seed scenario.Although there is evidence for this assumption (Baldassare et al. 2020), there is no consensus among simulations and semi-analytical models (Fontanot et al. 2015).However, under this assumption, we predict that laser and atom interferometers will be to place new and competitive constraints after their first years of observation, shedding light on the origin of SMBHs.

MODEL
We use the extended Press-Schechter formalism (Press & Schechter 1974;Bond et al. 1991;Lacey & Cole 1993) to estimate of the rate R h of coalescences of galactic halos of masses M 1,2 .Assuming conservatively that each of the halos includes at most one BH and that the BHs merge with the halos with probability p BH (Ellis et al. 2023b), the merger rate of BHs can be written as where p occ (m BH |M v , z, θ) is the occupation fraction of BHs of mass m BH in halos of mass M v at redshift z, and the mechanisms for SMBH formation are characterized by a set of parameters θ.We fix the value of the merging efficiency p BH to 0.84, which corresponds to the best fit of the SGWB measured by the NANOGrav collaboration (Ellis et al. 2023a).This fit was obtained including environmental effects on the binary evolution.Such effects are preferred by the fit but will not affect the binaries at the frequencies considered in this study.
We model the occupation fraction p occ (m BH |M v , z, θ) by a sum over different SMBH seed channels: where the log-normal distribution parameterizes the halo mass-BH mass relation and p j parametrizes the low-mass cut of the massive BH population.We compute the latter by using the observed halo mass-stellar mass relation At low redshifts, z ≲ 7, different SMBH seed scenarios are reflected in differences in the low-mass end of the occupation fraction.This is accounted for by the functions p j (m BH , θ j ) for which we consider the following parametric form: with the parameters θ j = (f j , m cut,j , w j ) where f j is the fraction of SMBHs produced by a given mechanism j, m cut,j characterizes a cut on the minimal SMBH mass, and w j is the spread with which the minimalmass cut is applied. 4We fix j f j = 1.The function p j is shown in the left panel of Fig. 1 for different values of m cut and w.In the following analysis, we consider two cases: (1) a one-component model where the SMBH population arises from one seed population and (2) a two-component model that includes a population arising from light seeds, e.g., BH remnants of Population-III stars, and a population arising from heavy seeds, e.g., BH nuclei of protogalaxies.We show an example of the occupation fraction p occ in the latter case in the right panel of Fig. 1. 5We emphasize that the above estimate of the BH merger rate is subject to various uncertainties.In particular, we neglect the additional delays related to the binary evolution following a halo merger (Hao et al. 2023) as well as possible difficulties for the SMBH to shrink to the galactic cores in dwarf galaxies (Dayal et al. 2020;Dunn et al. 2020), and we extrapolate the halo mass-BH mass relation to much lower masses than where it is currently measured.However, since our goal is mainly to illustrate and compare the capabilities of LISA, AION and AEDGE, a detailed discussion of overall uncertainties is beyond the scope of this work.

GW DATA ANALYSIS
For each specific SMBH formation model, characterized by the parameters θ, we generate n Monte Carlo (MC) realizations of the expected binary populations using Eq. ( 1).The number of binaries N j in any given MC realization follows a Poisson distribution whose mean is the expected number N (θ) of detectable binaries during one year of observation.The latter is given by (4) where SNR(x) is the signal-to-noise ratio of the GW signal from a binary with parameters x in an optimal source-detector system for a given GW detector, p det is the detection probability that accounts for the source location and orientation (Finn & Chernoff 1993;Gerosa et al. 2019), and SNR c = 8.
To characterize how well the parameters of the formation mechanism can be measured, we compute the likelihood for each of the MC realizations (Mandel et al. 2019;Hütsi et al. 2021): where The distributions P i (x i |x) account for the accuracy with which the parameters of the binary i can be measured.
We assume that P i (x i |x) follows a multivariate Gaussian distribution centered around the true values x i , and estimate the covariance matrix from the Fisher information, Σ kl (x i ) = Γ −1 kl (x i ).Finally, we compute the expected likelihood as and use that to estimate the expected accuracy at which the model parameters θ can be measured.
We express the optimal signal-to-noise ratio and the Fisher matrix Γ kl (x) of a signal as (Poisson & Will 1995) by defining the inner product of two complex functions a(f ) and b(f ) as where T denotes the observation time and S n (f ) is the noise power spectral density of the considered GW experiment.The partial derivatives in the Fisher matrix are taken with respect to the parameters of the template.We take into account the binary component masses m 1 and m 2 , its redshift z, the phase of the GW signals ϕ c and its coalescence time τ , but not the binary sky location or inclination, over which we simply average.To estimate the Fourier transform of the GW strain, h(x), we use the inspiral-merger-ringdown template (Ajith et al. 2008).

Binary population
We perform the analysis for LISA, AEDGE and AION-1km.As can be seen in the left panel of Fig. 2, LISA is expected to observe more binaries than AEDGE.For low-mass seed scenarios this difference is small but it grows with m cut implying that LISA will perform much better than AEDGE for heavy-seed scenarios.In particular, the expected number of binaries observable with AEDGE is less than one for heavy-seed scenarios with m cut > 10 5 M ⊙ .6For Fig. 2 we have fixed the width of the mass cut to w = 1, but we have checked that small changes to w do not alter significantly these results.Focusing on binaries that can be observed within 1 minute of the merger, the number of events for LISA is reduced to only ∼ 5 events for the light-seed case.In the heavyseed case, LISA will not see the last 1 minute of the binaries if m cut > 10 5 M ⊙ .On the other hand, AEDGE would detect most of the binaries until within 1 minute of the merger, except for those with M c ≲ 100M ⊙ .AION-1km has a similar range to AEDGE but, because it has less sensitivity, the number of events is decreased to around 10 if m cut ∼ 10 2 M ⊙ and further reduced to 1 event on average for 2 × 10 4 M ⊙ .
An explicit realization of the binary population in a light-seed scenario with (m cut = 10 2 M ⊙ ) is shown in the middle panel Fig. 2, and in a heavy-seed scenario with (m cut = 10 5 M ⊙ ) in the right panel of Fig. 2. We see that AEDGE and, to a lesser extent, AION-1km will probe IMBH binaries over a range of masses that are too light for LISA, covering the mass range covered by terrestrial laser interferometers from M c < 100M ⊙ to ∼ 10 4 M ⊙ .LISA as expected, will probe heavier binaries up to M c ∼ 10 7 M ⊙ .The types of events each experiment can detect is quite different.Considering the final stages within one minute of a merger, AEDGE binaries can explore the range M ∈ 10 2 − 10 4 M ⊙ out to redshifts z ∼ 7. LISA, on the other hand, observes only the last minutes of heavier binaries with M ∈ 10 4 − 10 6 M ⊙ and z ≲ 4. Finally, AION-1km will detect a handful of events for m cut < 10 4 M ⊙ , at low redshifts, z < 4, and mainly in the 10 3 − 10 4 M ⊙ mass range, mostly in the last moments before the merger.

Binary parameters
In Fig. 3 we display the accuracies with which LISA, AEDGE and AION-1km can measure the binary parameters for two representative choices of the binary component masses, a lighter binary with (m 1 , m 2 ) = (500, 300) M ⊙ (upper panels) and a heavier binary with (m 1 , m 2 ) = (10 4 , 5 × 10 4 ) M ⊙ (lower panels), as functions of the binary redshift z (left panels) and the time to merge τ (right panels).In general, the measurement accuracy of the binary parameters m 1 , m 2 and z are similar, and the measurement accuracy is best with AEDGE.The lighter binary enters the AION/AEDGE sensitivity band about a day before the merger.Conse-quently, the binary is not observable for AION/AEDGE if its coalescence time at the beginning of the experiment is longer than the observation time (T = 1 yr).LISA instead sees only the inspiralling part until less than a day from the merger and can see binaries whose coalescence time is τ ≲ 10 yr.The heavier binary is seen with AION/AEDGE for an even shorter time, but since the signal is strong the measurement accuracy is very good.In this case, also LISA sees the merger but the measurement with AEDGE is still more accurate unless the redshift of the binary is high, z ≳ 4, or its coalescence time is longer than the observation time.
For many binaries, the measurement accuracy with AEDGE or LISA is excellent.The probability distributions of the binary parameters can then be approximated by delta functions in the computation of the likelihood (5), so that the integrals over the binary parameters become trivial because the merger rate is essentially constant over such small parameter ranges.This reduces significantly the computational time for the likelihood.We adopt the delta-function approximation when all binary parameters are measured with better than 10% accuracy.

Seed scenarios
To estimate how accurately m cut could be measured with one-year data samples, we have generated n = 20 realizations of the detectable binary populations for a given value of m cut considering a single SMBH seed population.The left panel of Fig. 4 shows on the vertical axis the 95% CL ranges of m cut estimates for m cut ∈ [10 2 , 10 6 ] M ⊙ and taking a flat prior for log m cut over the range m cut ∈ [10 2 , 10 6 ] M ⊙ .The solid and dashed curves correspond to w = 1 and w = 2 and show that the measurement accuracy of m cut in these two 10 -4 10 -3 10 -2 0.1 1 10 10 2 10 3 10 4 10 -3 cases is very similar.We see that the seed mass is best recovered with LISA with a small uncertainty of a factor ∼ 2. For m cut ≲ ×10 3 M ⊙ the accuracy with AEDGE is similar to LISA, but the 95% CL range for AEDGE grows for heavier masses.When m cut ≳ 10 5 M ⊙ only a lower bound is obtained with AEDGE because the expected number of detectable events is less than one in the heavy-seed scenarios.On the other hand, already AION-1km could measure m cut ≲ 3 × 10 4 M ⊙ within an order of magnitude at 95% CL and could place a 95% CL lower bound of In the right panel of Fig. 4, we have considered the possibility that there are two different SMBH seed populations, a light one with m cut,1 = 100 M ⊙ and a heavy one with m cut,2 = 10 5 M ⊙ .We have again generated n = 20 realizations of the detectable binary populations for a given value of f 1 with f 2 = 1 − f 1 .In the right panel of Fig. 4 we show how accurately f 1 can be recovered at the 95% CL.We see that LISA could measure f 1 with an accuracy ∼ 10%, whereas AEDGE could measure f light with an accuracy ∼ 10 − 20%.AION-1km has a much bigger uncertainty but it would be precise enough that it can differentiate at the 95% CL a pure light-seed from a pure heavy-seed scenario.
For the right panel of Fig. 4 we have assumed delta function priors on m cut,1 and m cut,2 that might be theoretically motivated.Instead, in Fig. 5 consider a benchmark case with m 1 = 10 2 M ⊙ , m 2 = 10 5 M ⊙ and f 1 = f 2 = 0.5, and show the one-and twodimensional posteriors for their measurements with AION-1km (top panel), AEDGE (middle panel) and LISA (bottom panel) assuming the log-uniform priors 2 < log 10 (m cut,1 /M ⊙ ) < 3 and 3 < log 10 (m cut,2 /M ⊙ ) < 6 for the cutoff masses.For simplicity, we keep w 1 = w 2 = 1 fixed.We see that AION-1km provides a good estimate of m cut,1 , but provides only limited information on m cut,2 and f 1 .If we would take the prior m cut,2 ≳ 2 × 10 4 M ⊙ then we would reproduce the lower bound on f 1 with AION-1km seen in the right panel of Fig. 4. On the other hand, AEDGE provides good estimates of both m cut,1 and f 1 but also does not constrain m cut,2 significantly while LISA not only provides good estimates of both m cut,1 and f 1 but also constrains m cut,2 significantly.The marginalized posteriors of f 1 for AEDGE and LISA roughly match to the results shown in the right panel of Fig. 4.

CONCLUSIONS
We have described in this paper the capabilities of the planned space-borne laser interferometer LISA and the proposed atom interferometers AEDGE and AION-1km to observe mergers of intermediate-mass BHs, mea-sure their parameters, and discriminate between different seed scenarios for the assembly of SMBHs.We have considered the extended Press-Schechter to model the coalescences of galactic halos and estimate a rate for mergers of SMBHs that is compatible with the PTA signals for GWs in the nHz range.We have extrapolated this model to different SMBH seed scenarios by parametrizing the low mass cutoff of the massive BH population.Using this parametrization, we have estimated the possible rates for IMBH mergers, and assessed their detectability and measurability.
We have found that, although LISA has a high rate for observing the early infall stages of IMBH binaries for all the masses studied, this detector many binaries as the merger time approaches.Both AEDGE and AION-1km have higher rates than LISA for detections within one minute of the merger.We have shown that AEDGE has the best perspectives for detecting mergers of IMBHs weighing ≲ 10 4 M ⊙ whereas LISA has better perspectives for IMBHs weighing ≳ 10 5 M ⊙ .The better detection rates translate into smaller uncertainties in the measurements by AEDGE of binary parameters for IMBHs weighing ≲ 10 5 M ⊙ .
We have estimated the accuracy with which a lower cutoff on the BH seed mass, m cut could be extracted from the prospective GW data.We find that both LISA and AEDGE could determine m cut with precision ≲ 20% if m cut ≲ 10 4 M ⊙ , whereas LISA could determine m cut with better precision than AEDGE if m cut ≳ 10 4 M ⊙ .We also find that both LISA and AEDGE have interesting capabilities for distinguishing between scenarios with different mixtures of seeds with 10 2 and 10 5 M ⊙ .AION-1km could also provide some information, particularly in scenarios with a population of low-mass seeds.
Our results indicate that the space-borne laser interferometer LISA and atom interferometers AEDGE and AION-1km have interesting and complementary capabilities for measuring IMBH mergers and distinguishing between different seed scenarios for the assembly of SMBHs.We should emphasize that our study has been exploratory and should be complemented by an improved modelling of the SMBH seed scenarios and more detailed studies of the instrumental capabilities of GW interferometers.It would also be interesting to extend the analysis to assess the prospects for multimessenger observations and study the prospects for measuring higher-order multipoles of the GW signals that would allow for example for new probes of strong gravity.

Figure 1 .
Figure 1.Left panel: The low-mass cut on the massive BH population for different values of mcut, for w = 1 (solid) and w = 2 (dashed).Right panel: The halo mass-BH mass relation: The color coding shows the BH occupation fraction pocc as a function of the halo mass Mv and the BH mass mBH for a scenario with f1 = f2 = 0.5, w1 = w2 = 1, mcut,1 = 100M⊙ and mcut,2 = 10 5 M⊙.

Figure 2 .
Figure2.Left panel: The expected numbers of binaries detectable by AION-1km, AEDGE and LISA during a year of observation, as functions of mcut.The solid curves show all detectable binaries whereas the dotted curves show only those for which the last 1 minute of the merger is seen.Middle and right panels: Explicit examples of the detectable binaries for a light-seed and a heavy-seed scenario.The sizes of the dots are ∝ ln[SNR −1 ] with the minimum size corresponding to SNR = 10 4 and the maximal to SNR = 10.The darker dots correspond to binaries for which the last 1 minute of the merger is seen.