# On the single-event-based identification of primordial black hole mergers at
cosmological distances

Ken K. Y. Ng<EMAIL_ADDRESS>LIGO, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Shiqi Chen  LIGO,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA  Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Boris Goncharov  Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy
INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy  Ulyana
Dupletsa  Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy  INFN,
Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy  Ssohrab Borhanian
Institute for Gravitation and the Cosmos, Department of Physics, Pennsylvania
State University, University Park, PA, 16802, USA  Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, 07743, Jena, Germany  Marica
Branchesi  Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy  INFN,
Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy  Jan Harms  Gran
Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy  INFN, Laboratori
Nazionali del Gran Sasso, I-67100 Assergi, Italy  Michele Maggiore
Département de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai Ansermet, CH–1211 Genève 4, Switzerland  B. S.
Sathyaprakash  Institute for Gravitation and the Cosmos, Department of
Physics, Pennsylvania State University, University Park, PA, 16802, USA
Department of Astronomy & Astrophysics, Pennsylvania State University,
University Park, PA, 16802, USA  School of Physics and Astronomy, Cardiff
University, Cardiff, UK, CF24 3AA  Salvatore Vitale  LIGO, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Kavli Institute
for Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

The existence of primordial black holes (PBHs), which may form from the
collapse of matter overdensities shortly after the Big Bang, is still under
debate. Among the potential signatures of PBHs are gravitational waves (GWs)
emitted from binary black hole (BBH) mergers at redshifts $z\gtrsim 30$, where
the formation of astrophysical black holes is unlikely. Future ground-based GW
detectors, Cosmic Explorer and Einstein Telescope, will be able to observe
equal-mass BBH mergers with total mass of $\mathcal{O}(10-100)~{}M_{\odot}$ at
such distances. In this work, we investigate whether the redshift measurement
of a single BBH source can be precise enough to establish its primordial
origin. We simulate BBHs of different masses, mass ratios and orbital
orientations. We show that for BBHs with total masses between $20~{}M_{\odot}$
and $40~{}M_{\odot}$ merging at $z\geq 40$ one can infer $z>30$ at up to 97%
credibility, with a network of one Einstein Telescope, one 40-km Cosmic
Explorer in the US and one 20-km Cosmic Explorer in Australia. A smaller
network made of one Einstein Telescope and one 40-km Cosmic Explorer in the US
measures $z>30$ at larger than 90% credibility for roughly half of the sources
than the larger network. We then assess the dependence of this result on the
Bayesian redshift priors used for the analysis, specifically on the relative
abundance of the BBH mergers originated from the first stars, and the
primordial BBH mergers.

Introduction – Formation of primordial black holes (PBHs) was suggested more
than five decades ago [1, 2, 3], but these hypothetical compact objects still
elude discovery. Unlike astrophysical black holes (ABHs) – which are stellar
remnants – PBHs are formed by the direct collapse of matter overdensities in
the early Universe [4, 5, 6] (also see Refs. [7, 8, 9, 10] for reviews). While
there are significant uncertainties in their mass spectrum, PBH may occur in
the range $\mathcal{O}(1$–$100)~{}M_{\odot}$ [2, 11, 12]. Stellar-mass PBHs
could form binaries [13, 14] that merge and emit gravitational waves (GWs)
detectable by current GW detectors, including LIGO, Virgo and KAGRA (LVK) [15,
16, 17], and leave a unique imprint on the mass spectrum and redshift
evolution in the observable population of binary black hole (BBH) mergers [18,
19, 12, 20, 21, 22, 23, 24]. Efforts have been made to test if a fraction of
the BBHs in the LVK’s second catalog [25] could be of primordial origin, by
analysing the population properties of BBH mergers detected thus far [26, 27,
28, 29, 30]. However, even at design sensitivity, the horizon of current GW
detectors will be limited to redshifts of $z\lesssim 3$ at most [31].
Interpreting these “local” observations, with the aim of establishing the
presence of a PBH sub-population, requires precise knowledge of the ABH
population, which is dominant at low redshifts and acts as an “astrophysical
foreground” [29]. This is challenging, as there exist significant
uncertainties on the properties of BBHs formed in different astrophysical
environments, such as galactic fields [32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42], dense star clusters [43, 44, 45, 46, 47, 48, 49, 50, 46, 47, 48, 44],
active galactic nuclei [51, 52, 53, 54, 55, 56, 57, 58, 59], or from the
collapse of Population III (Pop III) stars [60, 61, 62, 63].

One can ascertain the primordial origin of black holes by detecting mergers at
redshifts so high that ABHs could not have had the time to form and merge yet.
A plausible lower bound for this redshift would be $z\sim 30$ (see discussion
below). Therefore, measuring the redshift of a BBH merging at redshift larger
than $30$ would be a clear hint of the existence of PBHs [64, 65].

Proposed next-generation, ground-based GW detectors such as the Cosmic
Explorer (CE) [66, 67, 68] and the Einstein Telescope (ET) [69, 70] can detect
BBH mergers at $z>10$ and above [31]. However, being able to _detect_ a source
merging at redshift larger than $30$ does not automatically imply being able
to prove that the true redshift was above some threshold. The purpose of this
Letter is to systematically study how well next-generation GW detectors can
measure the redshift of distant BBHs.

Simulations – Given the significant uncertainty in the mass spectrum of PBHs,
we consider a range of values which would lead to detectable GW emission. We
simulate BBHs merging at redshifts of $z=10,20,30,40$ and $50$. The total mass
in the source frame is chosen to be $M_{\rm tot}=5$, 10, 20, 40, 80, 160, and
250 $M_{\odot}$, with mass ratio $q=1$, 2, 3, 4 and 5 (where $q\equiv
m_{1}/m_{2}$ for $m_{1}>m_{2}$) and four values of the orbital inclination
$\iota=0$ (face-on), $\pi/6$, $\pi/3$, and $\pi/2$ (edge-on). The simulated
BBHs are non-spinning, as it is expected that PBHs are born with negligible
spins [71, 72] and may be spun-up by accreting materials later in their lives
[73, 24, 23, 27]. However, we do _not_ assume that spins are zero when
estimating the source parameters and instead allow for generic spin-
precession. For each of these 700 sources, the sky location and polarization
angles are chosen to maximize the source’s signal-to-noise ratio (SNR).

In order to measure a source’s distance (and hence redshift 111Throughout this
study, we use a $\Lambda$CDM cosmology based on the Planck 2018 results [106]
to convert luminosity distance into redshift.), one needs to disentangle the
two GW polarizations. Thus, we only consider _networks_ of non-collocated
next-generation observatories as opposed to single-site detectors. We work
with two different detector networks: (i) a 40-km CE in the United States and
an ET in Europe (CE-ET), and (ii) CE-ET with an additional 20-km CE in
Australia (CE-CES-ET). We only analyze sources whose network SNRs are larger
than 12. Generally speaking, BBHs with $q\geq 5$, $M_{\rm tot}\geq
160M_{\odot}$ or $M_{\rm tot}\leq 5M_{\odot}$ can only be detected up to
$z\sim 20$.

We obtain posterior probability densities with bilby [75, 76, 77]. The
inference is made with a zero-noise realization [78], since we are only
interested in the uncertainty caused by limited SNR, and do not want to be
prone to offsets potentially caused by large Gaussian fluctuations [79]. We
employ the IMRPhenomXPHM waveform family, which accounts for the effects of
spin precession and higher-order modes (HoMs) [80], both to create the
simulated waveforms and to calculate the likelihood in the sampler. It is
necessary to include HoMs in our analysis as they are important for systems
with large detector-frame masses ($\gtrsim 100M_{\odot}$) and large mass
ratios. We expect HoMs to be particularly relevant for the analyses we
present, as they help break the distance-inclination degeneracy characteristic
of the dominant (2,2) multipole mode [81, 82]. We find it more efficient to
sample the parameter space with uniform priors in the detector-frame total
mass, $M_{\rm tot}(1+z)$, and the mass ratio, $q$. The posteriors obtained
this way are then reweighed into uniform priors in the source-frame primary
mass, $m_{1}$, and the inverse mass ratio $1/q$ (which is between 0 and 1).
The default prior on redshift is uniform in the source-frame differential
comoving volume, $p_{0}(z)\propto\frac{dV_{c}}{dz}\frac{1}{1+z}$, but we will
explore other options below. We used isotropic priors for the sky position,
the orbital and spin orientations. Uniform priors were used for the arrival
time, phase of the signal at the time of arrival, and the spin magnitude.
Unless otherwise specified, in what follows we quote uncertainties at 95%
credible intervals.

Redshift uncertainties – Figure 1 shows the redshift posteriors for a BBH
system with $(M_{\rm tot},q,\iota)=(40M_{\odot},1,\pi/6)$ located at different
redshifts (given in the $x$-axis), as observed by CE-ET (red) and CE-CES-ET
(blue). To increase the resolution along the $y$-axis, we have offset each
posterior by the true redshift value. The redshift uncertainty increases with
the true redshift, from $\Delta z\sim 5$ at $z_{\mathrm{true}}=10$ to $\Delta
z\sim 30$ at $z_{\mathrm{true}}=50$ for a CE-CES-ET network.

Figure 1: Redshift posteriors (offset by the true values) for sources with
$(M_{\rm tot},q,\iota)=(40M_{\odot},1,\pi/6)$ at 5 different redshifts
observed by CE-ET (red) and CE-CES-ET (blue). The solid horizontal lines show
the 95% credible intervals, whereas the dashed lines mark $z_{\mathrm{true}}$.

The roughly linear increase of the uncertainty with redshift is not just due
to the reduction of the signal’s amplitude, as it happens for the sources
discovered by advanced detectors. For any frequency in the inspiral phase of
the waveform, the Fourier amplitude is proportional to
${\left[(1+z)\mathcal{M}_{c}\right]^{5/6}}{d_{L}}^{-1}$, where
$\mathcal{M}_{c}$ is the source-frame chirp mass and $d_{L}$ is the luminosity
distance. At large redshifts $d_{L}\propto(1+z).$ Thus, for a given source-
frame $\mathcal{M}_{c}$, the inspiral Fourier amplitude of sources at $z\gg 1$
only falls off as $(1+z)^{-1/6}$. However, the loss in SNR at higher redshifts
is greater because frequencies are redshifted by a factor of $(1+z)$ where the
detectors’ sensitivity might be poorer but also because signals have smaller
bandwidth in the detectors’ sensitivity range. These effects add up to yield
the trend observed in Fig. 1.

We also note that the network without CES yields uncertainties which are up to
$\sim 10\%$ larger than the three-detector network. This happens because an
extra detector improves the resolution of the GW polarizations.

Next, we discuss how the redshift uncertainty depends on the total mass of the
system. In Fig. 2 we show the redshift posteriors of BBHs of increasing
$M_{\rm tot}$, with $z_{\mathrm{true}}=40$, $q=1$ and $\iota=0$. The redshift
uncertainty obtained by CE-CES-ET first decreases from $\Delta z\sim 20$ for
$M_{\rm tot}=10M_{\odot}$ to $\Delta z\sim 15$ for $M_{\rm tot}=40M_{\odot}$,
then increases to $\Delta z\sim 25$ for $M_{\rm tot}=80M_{\odot}$ (this plot
does not feature a system with $M_{\rm tot}=5$ and $160~{}M_{\odot}$ as its
SNR is below the threshold of 12). The non-linear trend apparent in this plot
is the result of two competing effects. At first, as the mass increases, so do
the signal amplitude and the SNR. In this regime, the uncertainty decreases
with $M_{\rm tot}$. However, when $M_{\rm tot}$ increases further, signals
sweep less of the detectors’ sensitive bandwidth. In this regime, the
uncertainties increase with $M_{\rm tot}$.

Figure 2: Redshift posteriors for sources with $(q,\iota,z)=(1,0,40)$ at 4
different total masses observed by CE-ET (red) and CE-CES-ET (blue). The solid
horizontal lines show the 95% credible intervals, whereas the dashed line
marks $z_{\mathrm{true}}$. Only sources with $\rm SNR\geq 12$ are included.

We also notice an offset between the true redshift and the maximum a
posteriori redshift of each source. There are two reasons for this offset. The
default redshift prior goes as $p_{0}(z)\sim(1+z)^{-5/2}$ in the matter-
dominated regime, $1\lesssim z\lesssim 1000$, hence favoring lower redshifts.
Furthermore, the true inclination of these sources is $0$, which implies that
underestimating the redshift can be compensated for by measuring inclination
angles closer to edge-on. The opposite is obviously not possible since the
true inclination is at the edge of its physically allowed prior. This
asymmetry makes it easier to underestimate the redshift but more difficult to
overestimate it. This is why this systematic offset was not observed in Fig.
1, where the true inclination was $\pi/6$. The effect of priors is discussed
more extensively below.

Can single-event-based PBH identification be unambiguous? – With these results
at hand, we now address the key question of this study: can one pinpoint the
redshift of merging BBHs to be above some threshold value? The answer will
clearly depend on which threshold is used. While the redshift at which the
first stars were born is not precisely known, theoretical calculations and
cosmological simulations suggest that the first stars might have been born at
redshifts smaller than $z\sim 40$ [83, 84, 85, 86] 222Although, we note that
there are studies suggesting an earlier $(z\gtrsim 50)$ [107] or a later
$(z\lesssim 20)$ [108] formation of Pop III stars.. Meanwhile, population
synthesis models indicate that the characteristic time delay between formation
and merger for Pop III stars is $\mathcal{O}(10)~{}\rm Myr$ [60, 61, 62, 63,
88, 89, 90, 91, 92]. This implies that the redshift at which Pop III remnant
BBHs merge is below $z\sim 30$. On the other hand, stellar-mass PBHs are
expected to have formed at $z\gg 1000$, much earlier than the recombination
epoch, and have been merging with one another since then [22]. The PBH merger
rate density therefore increases monotonically with redshift [22], unlike that
of ABHs.

Based on the above, we choose a critical redshift $z_{\mathrm{crit}}=30$,
above which no astrophysical BBHs are expected to merge. We define the
probability of primordial origin, $P_{\text{p}}$ as the fraction of the
redshift posterior with $z\geq z_{\mathrm{crit}}$. In the following, we only
consider the measurements made by CE-CES-ET which are always better than those
made by CE-ET.

Table 1: Parameters of the simulations that result in a probability of a primordial origin $P_{\text{p}}\geq 0.9$, assuming the default prior $p_{0}(z)$. We also show the individual SNR in CE, $\rho_{\rm CE}$, SNR in CES, $\rho_{\rm CES}$, SNR in ET, $\rho_{\rm ET}$, as well as the network SNR, $\rho_{\rm net}$.  $M_{\rm tot}$ | $q$ | $\iota$ | $z$ | $\rho_{\rm CE}$ | $\rho_{\rm CES}$ | $\rho_{\rm ET}$ | $\rho_{\rm net}$ | $P_{p}$
---|---|---|---|---|---|---|---|---
20 | 1 | 0 | 50 | 16.6 | 7.2 | 5.9 | 19.1 | 0.93
20 | 1 | $\pi/6$ | 50 | 14.0 | 6.3 | 5.2 | 16.2 | 0.94
20 | 2 | 0 | 50 | 13.6 | 5.8 | 5.2 | 15.7 | 0.95
20 | 2 | $\pi/6$ | 50 | 11.6 | 5.1 | 4.5 | 13.5 | 0.91
20 | 3 | 0 | 40 | 13.0 | 5.7 | 4.5 | 14.9 | 0.92
40 | 1 | $\pi/6$ | 40 | 7.4 | 6.9 | 16.8 | 19.6 | 0.91
40 | 1 | $\pi/3$ | 40 | 6.0 | 4.6 | 9.5 | 12.1 | 0.97
40 | 1 | 0 | 50 | 3.0 | 4.2 | 17.8 | 18.5 | 0.94
40 | 1 | $\pi/6$ | 50 | 2.6 | 3.5 | 15.1 | 15.7 | 0.96
40 | 2 | 0 | 40 | 5.6 | 6.0 | 17.2 | 19.0 | 0.92
40 | 2 | 0 | 50 | 2.1 | 3.0 | 14.5 | 15.0 | 0.93
40 | 3 | 0 | 40 | 3.2 | 4.0 | 13.7 | 14.7 | 0.91

In Table 1, we list the BBH sources for which $P_{\text{p}}\geq 0.9$, and the
corresponding values of $P_{\text{p}}$. We find that only sources with $z\geq
40$ and $M_{\rm tot}=20M_{\odot}$ or $40M_{\odot}$ achieve $P_{\text{p}}\geq
0.9$. We stress that the highest $P_{\text{p}}$’s are not reached by face-on
sources, despite the fact that they have the largest SNRs. This is because the
amplitude of HoMs is smallest for face-on sources, which will suffer the most
from the distance-inclination degeneracy, and hence have larger uncertainties.
We will explore this trade-off further in a forthcoming paper.

Next, we want to explore the dependence of these results on the redshift
prior. The results presented thus far used our default prior for the redshift,
uniform in co-moving volume and source-frame time, given above. One could
instead decide to use a prior informed by reasonable merger rates for ABHs and
PBHs. In this case, the analysis of $P_{\text{p}}$ answers the following
question: given some expected distribution of merger rates of PBHs and ABHs,
what is the probability that _this_ system is primordial? To answer
quantitatively, we adopt for the PBHs the merger rate density $\dot{n}_{\rm
PBH}$, from Refs. [19, 22, 27] $\dot{n}_{\rm
PBH}(z)\propto\left(\frac{t(z)}{t(0)}\right)^{-34/37}$, and that for the Pop
III BBH mergers, $\dot{n}_{\rm III}$, from Refs. [63, 93],

$\displaystyle\dot{n}_{\rm III}(z)\propto\begin{cases}\frac{e^{a_{\rm
III}(z-z_{\rm III})}}{b_{\rm III}+a_{\rm III}e^{(a_{\rm III}+b_{\rm
III})(z-z_{\rm III})}}&\mathrm{if}\,z<z_{\mathrm{crit}}\\\
0&\mathrm{otherwise}\end{cases},$ (1)

with $t(z)$ the age of the Universe at $z$, and $(a_{\rm III},b_{\rm
III},z_{\rm III})=(0.66,0.3,11.6)$ [93].

Then, we construct a mixture model for the merger rate based redshift prior,
$p_{\rm tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})$,

$\displaystyle p_{\rm
tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})\propto\left[f^{\mathrm{PBH}}_{\mathrm{III}}\frac{\dot{n}_{\rm
PBH}(z)}{\dot{n}_{\rm PBH}(z_{\mathrm{crit}})}+\frac{\dot{n}_{\rm
III}(z)}{\dot{n}_{\rm
III}(z_{\mathrm{crit}})}\right]\frac{dV_{c}}{dz}\frac{1}{1+z},$

where $f^{\mathrm{PBH}}_{\mathrm{III}}$ represents the ratio of the merger
rate contributed by PBH mergers to that contributed by Pop III BBH mergers at
$z_{\mathrm{crit}}$: $f^{\mathrm{PBH}}_{\mathrm{III}}\equiv{\dot{n}_{\rm
PBH}(z_{\mathrm{crit}})}/{\dot{n}_{\rm III}(z_{\mathrm{crit}})}$. We plot
$p_{\rm tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})$ for 5 different value of
$f^{\mathrm{PBH}}_{\mathrm{III}}$ in the Supplementary Material.

We then re-analyze the source with the largest $P_{\text{p}}$ of Table 1,
namely, $(M_{\rm tot},q,\iota,z)=(40M_{\odot},1,\pi/3,40)$, by applying the
rate-based redshift prior $p_{\rm tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})$
instead of the default prior $p_{0}(z)$. The redshift posteriors for different
$f^{\mathrm{PBH}}_{\mathrm{III}}$ are shown in Fig. 3. When
$f^{\mathrm{PBH}}_{\mathrm{III}}$ decreases below 1, the posterior becomes
bimodal with a low-redshift peak around $z\approx 20$, which becomes the
dominating peak for $f^{\mathrm{PBH}}_{\mathrm{III}}\lesssim 0.1$. We tabulate
the values of $P_{\text{p}}$ for the same system, calculated with the
different priors in Table 2. The value of $P_{\text{p}}$ drops below $0.95$
for $f^{\mathrm{PBH}}_{\mathrm{III}}\leq 1$. This seems to suggest that unless
one believes a priori that the relative fraction between PBH mergers and Pop
III BBH mergers at $z_{\mathrm{crit}}$ is at least unity, the probability that
a specific source is primordial will be low.

However, not all priors yield comparable Bayesian evidences. One can thus
calculate the Bayes factor $\mathcal{B}^{\mathrm{P}}_{0}=Z_{\rm P}/Z_{0}$,
where $Z_{\rm P}$ and $Z_{0}$ are the single-event evidences calculated with
the rate-based prior and the default prior, respectively [94, 95, 96]. Values
of $\mathcal{B}^{\mathrm{P}}_{0}>1$ imply that the default prior is
disfavoured. As shown in Table 2, the value of $\mathcal{B}^{\mathrm{P}}_{0}$
decreases from $\sim 2$ to $\sim 3\times 10^{-3}$ with
$f^{\mathrm{PBH}}_{\mathrm{III}}$, meaning that rate-based priors with small
PBH fraction $f^{\mathrm{PBH}}_{\mathrm{III}}\lesssim 0.1$ are strongly
disfavored when comparing to the rate based prior with
$f^{\mathrm{PBH}}_{\mathrm{III}}\gtrsim 1$. Therefore, while an a-priori
belief that the fraction of PBHs is low will move the redshift posterior to
lower values, it will also yield a low Bayes factor that disfavors the model
with a smaller $f^{\mathrm{PBH}}_{\mathrm{III}}$.

Figure 3: Posteriors of the system $(M_{\rm tot},q,\iota,z)=(40M_{\odot},1,\pi/3,40)$, using $p_{\rm tot}(z)$ evaluated at $f^{\mathrm{PBH}}_{\mathrm{III}}=100$ (blue), 10 (orange), 1 (green), 0.1 (red), and 0.01 (purple). The black solid line shows the default posterior. The black dashed line indicates $z_{\mathrm{true}}=40$, and the grey area indicates the astrophysical region $z<z_{\mathrm{crit}}$.  Table 2: Probabilities of a primordial origin $P_{\text{p}}$ of the system $(M_{\rm tot},q,\iota,z)=(40M_{\odot},1,\pi/3,40)$ and the Bayes factor $\mathcal{B}^{\mathrm{P}}_{0}$ against the default prior calculated with the rate based priors $p_{\rm tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})$ for different PBH-to-Pop III merger rate ratios at $z_{\mathrm{crit}}$, $f^{\mathrm{PBH}}_{\mathrm{III}}$. The value of $P_{\rm p}$ calculated with default prior $p_{0}(z)$ is shown for comparison. $f^{\mathrm{PBH}}_{\mathrm{III}}$ | $P_{\text{p}}$ | $\mathcal{B}^{\mathrm{P}}_{0}$
---|---|---
100 | 0.98 | 1.93
10 | 0.98 | 0.40
1 | 0.88 | 0.035
0.1 | 0.49 | 0.0030
0.01 | 0.11 | 0.0028
Default | 0.97 | 1

Discussion – In this Letter, we have simulated BBHs merging at $z\geq 10$ and
quantified the uncertainties in their redshifts as measured by networks of
next-generation ground-based gravitational-wave detectors. We have used full
Bayesian parameter estimation, and a waveform model which accounts for both
higher-order modes and spin precession. The relative redshift uncertainties
are larger than $\sim 10\%$ at $z\geq 10$, even for a three-detector network
CE-CES-ET. Assuming that no astrophysical BBHs can merge above
$z_{\mathrm{crit}}=30$ (see Supplementary Material for a lower threshold), we
found that the typical redshift measurement is not precise enough to conclude
with certainty that a single source is of primordial origin. Among the systems
we simulated, the ones for which one can establish the primordial nature more
strongly have $M_{\rm tot}=20M_{\odot}$ or $40M_{\odot}$ at
$z_{\mathrm{true}}\geq 40$. We find 12 such systems for which
$90\%<p(z>z_{\mathrm{crit}})<97\%$ assuming a uniform prior in source-frame
differential comoving volume and using a CE-CES-ET network. With a smaller CE-
ET network, only 3 sources have $90\%<p(z>z_{\mathrm{crit}})$. We have also
verified that with a single ET observatory one can reach
$p(z>z_{\mathrm{crit}})\sim 70\%$. With a smaller set of targeted simulations,
we have verified that with a single CE observatory one cannot set significant
constraints. This is expected as the capabilities of a single L-shaped
detector to measure distances outside of a network are limited.

Next, we have shown how the redshift measurement depends significantly on the
redshift prior used for the analysis. In particular, we have considered a
prior informed by population synthesis and theoretical models and introduced a
mixture model that allows for both ABHs and PBHs. As one would expect, the
posterior probability of primordial origin for a specific event decreases if
the assumed prior ratio of PBH merger rate to Pop III BBH merger rate is
small. However, the Bayes factor between the mixture model and the constant
comoving density model also decreases. Claims based on individual events would
thus benefit from better knowledge of the properties of Pop III stars and
their remnants, notably the highest rate at which they merge. Forthcoming
facilities such as the JWST, the Roman space telescope or Euclid are expected
to probe the properties of Pop III stars by accessing Pop III galaxies in
blind surveys [97]. More information will be yielded by instrument such as
SPHEREx, by precisely measuring the near infrared background [98]; and mission
concepts like THESEUS, by detecting the most distant long gamma-ray burst
[99].

While an event-by-event basis identification of PBH binaries might be
challenging, one can find evidence for PBHs in the whole dataset. There are
two possible methods to perform such an analysis. First, the subthreshold
events from PBH mergers may contribute to a distinctive stochastic GW
background, whose amplitude is directly related to the rate of PBHs mergers
[100]. Second, one may use multiple resolvable sources at high redshifts and
combine their redshift measurements through hierarchical Bayesian analysis.
This also allows for detailed modelling on the features of each subpopulation
[101, 102, 103, 104, 105]. We will explore this avenue in a future paper.

Acknowledgments – We would like to thank E. Berti, M. Evans, G. Franciolini,
P. Pani, M. Punturo, A. Riotto for suggestions and comments. KKYN and SV are
supported by the NSF through the award PHY-1836814. SC is supported by the
Undergraduate Research Opportunities Program of Massachusetts Institute of
Technology. The work of MM is supported by the Swiss National Science
Foundation and by the SwissMap National Center for Competence in Research. MB
acknowledges support from the European Union’s Horizon 2020 Programme under
the AHEAD2020 project (grant agreement n. 871158). BSS is supported in part by
NSF Grant No. PHY-1836779, AST-2006384 and PHY- 2012083. SB is supported by
NSF Grant No.PHY-1836779. BG is supported by the Italian Ministry of
Education, University and Research within the PRIN 2017 Research Program
Framework, n. 2017SYRTCN.

## References

  * Zel’dovich and Novikov [1966] Y. B. Zel’dovich and I. D. Novikov, The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model, AZh 43, 758 (1966).
  * Carr and Hawking [1974] B. J. Carr and S. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168, 399 (1974).
  * Hawking [1971] S. Hawking, Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc. 152, 75 (1971).
  * Ivanov  _et al._ [1994] P. Ivanov, P. Naselsky, and I. Novikov, Inflation and primordial black holes as dark matter, Phys. Rev. D 50, 7173 (1994).
  * Garcia-Bellido  _et al._ [1996] J. Garcia-Bellido, A. D. Linde, and D. Wands, Density perturbations and black hole formation in hybrid inflation, Phys. Rev. D 54, 6040 (1996), arXiv:astro-ph/9605094 .
  * Ivanov [1998] P. Ivanov, Nonlinear metric perturbations and production of primordial black holes, Phys. Rev. D 57, 7145 (1998), arXiv:astro-ph/9708224 .
  * Polnarev and Khlopov [1985] A. G. Polnarev and M. Y. Khlopov, COSMOLOGY, PRIMORDIAL BLACK HOLES, AND SUPERMASSIVE PARTICLES, Sov. Phys. Usp. 28, 213 (1985).
  * Khlopov [2010] M. Y. Khlopov, Primordial Black Holes, Res. Astron. Astrophys. 10, 495 (2010), arXiv:0801.0116 [astro-ph] .
  * Sasaki  _et al._ [2016] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914, Phys. Rev. Lett. 117, 061101 (2016), [Erratum: Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 [astro-ph.CO] .
  * Green and Kavanagh [2021] A. M. Green and B. J. Kavanagh, Primordial Black Holes as a dark matter candidate, J. Phys. G 48, 043001 (2021), arXiv:2007.10722 [astro-ph.CO] .
  * Carr  _et al._ [2017] B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen, and H. Veermäe, Primordial black hole constraints for extended mass functions, Phys. Rev. D 96, 023514 (2017), arXiv:1705.05567 [astro-ph.CO] .
  * Ali-Haïmoud  _et al._ [2017] Y. Ali-Haïmoud, E. D. Kovetz, and M. Kamionkowski, Merger rate of primordial black-hole binaries, Phys. Rev. D 96, 123523 (2017), arXiv:1709.06576 [astro-ph.CO] .
  * Nakamura  _et al._ [1997] T. Nakamura, M. Sasaki, T. Tanaka, and K. S. Thorne, Gravitational waves from coalescing black hole MACHO binaries, Astrophys. J. Lett. 487, L139 (1997), arXiv:astro-ph/9708060 .
  * Ioka  _et al._ [1998] K. Ioka, T. Chiba, T. Tanaka, and T. Nakamura, Black hole binary formation in the expanding universe: Three body problem approximation, Phys. Rev. D 58, 063003 (1998), arXiv:astro-ph/9807018 .
  * Aasi  _et al._ [2015] J. Aasi _et al._ (LIGO Scientific), Advanced LIGO, Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * Acernese  _et al._ [2015] F. Acernese _et al._ (VIRGO), Advanced Virgo: a second-generation interferometric gravitational wave detector, Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc] .
  * Aso  _et al._ [2013] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto (KAGRA), Interferometer design of the KAGRA gravitational wave detector, Phys. Rev. D 88, 043007 (2013), arXiv:1306.6747 [gr-qc] .
  * Clesse and García-Bellido [2017] S. Clesse and J. García-Bellido, The clustering of massive Primordial Black Holes as Dark Matter: measuring their mass distribution with Advanced LIGO, Phys. Dark Univ. 15, 142 (2017), arXiv:1603.05234 [astro-ph.CO] .
  * Raidal  _et al._ [2017] M. Raidal, V. Vaskonen, and H. Veermäe, Gravitational Waves from Primordial Black Hole Mergers, JCAP 09, 037, arXiv:1707.01480 [astro-ph.CO] .
  * Belotsky  _et al._ [2019] K. M. Belotsky, V. I. Dokuchaev, Y. N. Eroshenko, E. A. Esipova, M. Y. Khlopov, L. A. Khromykh, A. A. Kirillov, V. V. Nikulin, S. G. Rubin, and I. V. Svadkovsky, Clusters of primordial black holes, Eur. Phys. J. C 79, 246 (2019), arXiv:1807.06590 [astro-ph.CO] .
  * Chen and Huang [2018] Z.-C. Chen and Q.-G. Huang, Merger Rate Distribution of Primordial-Black-Hole Binaries, Astrophys. J. 864, 61 (2018), arXiv:1801.10327 [astro-ph.CO] .
  * Raidal  _et al._ [2019] M. Raidal, C. Spethmann, V. Vaskonen, and H. Veermäe, Formation and Evolution of Primordial Black Hole Binaries in the Early Universe, JCAP 02, 018, arXiv:1812.01930 [astro-ph.CO] .
  * De Luca  _et al._ [2020a] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Constraints on Primordial Black Holes: the Importance of Accretion, Phys. Rev. D 102, 043505 (2020a), arXiv:2003.12589 [astro-ph.CO] .
  * De Luca  _et al._ [2020b] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, The evolution of primordial black holes and their final observable spins, JCAP 04, 052, arXiv:2003.02778 [astro-ph.CO] .
  * Abbott  _et al._ [2020] R. Abbott _et al._ (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, arXiv e-print (2020), arXiv:2010.14527 [gr-qc] .
  * Wong  _et al._ [2021] K. W. K. Wong, G. Franciolini, V. De Luca, V. Baibhav, E. Berti, P. Pani, and A. Riotto, Constraining the primordial black hole scenario with Bayesian inference and machine learning: the GWTC-2 gravitational wave catalog, Phys. Rev. D 103, 023026 (2021), arXiv:2011.01865 [gr-qc] .
  * De Luca  _et al._ [2020c] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Primordial Black Holes Confront LIGO/Virgo data: Current situation, JCAP 06, 044, arXiv:2005.05641 [astro-ph.CO] .
  * Hütsi  _et al._ [2021] G. Hütsi, M. Raidal, V. Vaskonen, and H. Veermäe, Two populations of LIGO-Virgo black holes, JCAP 03, 068, arXiv:2012.02786 [astro-ph.CO] .
  * Franciolini  _et al._ [2021] G. Franciolini, V. Baibhav, V. De Luca, K. K. Y. Ng, K. W. K. Wong, E. Berti, P. Pani, A. Riotto, and S. Vitale, Quantifying the evidence for primordial black holes in LIGO/Virgo gravitational-wave data, arXiv (2021), arXiv:2105.03349 [gr-qc] .
  * Mukherjee and Silk [2021] S. Mukherjee and J. Silk, Can we distinguish astrophysical from primordial black holes via the stochastic gravitational wave background?, Mon. Not. Roy. Astron. Soc. 506, 3977 (2021), arXiv:2105.11139 [gr-qc] .
  * Hall and Evans [2019] E. D. Hall and M. Evans, Metrics for next-generation gravitational-wave detectors, Class. Quant. Grav. 36, 225002 (2019), arXiv:1902.09485 [astro-ph.IM] .
  * O’Shaughnessy  _et al._ [2017] R. O’Shaughnessy, J. Bellovary, A. Brooks, S. Shen, F. Governato, and C. Christensen, The effects of host galaxy properties on merging compact binaries detectable by LIGO, Mon. Not. Roy. Astron. Soc. 464, 2831 (2017), arXiv:1609.06715 [astro-ph.GA] .
  * Dominik  _et al._ [2012] M. Dominik, K. Belczynski, C. Fryer, D. Holz, E. Berti, T. Bulik, I. Mandel, and R. O’Shaughnessy, Double Compact Objects I: The Significance of the Common Envelope on Merger Rates, Astrophys. J. 759, 52 (2012), arXiv:1202.4901 [astro-ph.HE] .
  * Dominik  _et al._ [2013] M. Dominik, K. Belczynski, C. Fryer, D. E. Holz, E. Berti, T. Bulik, I. Mandel, and R. O’Shaughnessy, Double Compact Objects II: Cosmological Merger Rates, Astrophys. J. 779, 72 (2013), arXiv:1308.1546 [astro-ph.HE] .
  * Dominik  _et al._ [2015] M. Dominik, E. Berti, R. O’Shaughnessy, I. Mandel, K. Belczynski, C. Fryer, D. E. Holz, T. Bulik, and F. Pannarale, Double Compact Objects III: Gravitational Wave Detection Rates, Astrophys. J. 806, 263 (2015), arXiv:1405.7016 [astro-ph.HE] .
  * de Mink and Belczynski [2015] S. de Mink and K. Belczynski, Merger rates of double neutron stars and stellar origin black holes: The Impact of Initial Conditions on Binary Evolution Predictions, Astrophys. J. 814, 58 (2015), arXiv:1506.03573 [astro-ph.HE] .
  * Belczynski  _et al._ [2016] K. Belczynski, D. E. Holz, T. Bulik, and R. O’Shaughnessy, The first gravitational-wave source from the isolated evolution of two 40-100 Msun stars, Nature 534, 512 (2016), arXiv:1602.04531 [astro-ph.HE] .
  * Stevenson  _et al._ [2017] S. Stevenson, A. Vigna-Gómez, I. Mandel, J. W. Barrett, C. J. Neijssel, D. Perkins, and S. E. de Mink, Formation of the first three gravitational-wave observations through isolated binary evolution, Nature Commun. 8, 14906 (2017), arXiv:1704.01352 [astro-ph.HE] .
  * Mapelli  _et al._ [2019] M. Mapelli, N. Giacobbo, F. Santoliquido, and M. C. Artale, The properties of merging black holes and neutron stars across cosmic time, Mon. Not. Roy. Astron. Soc. 487, 2 (2019), arXiv:1902.01419 [astro-ph.HE] .
  * Breivik  _et al._ [2020] K. Breivik _et al._ , COSMIC Variance in Binary Population Synthesis, Astrophys. J. 898, 71 (2020), arXiv:1911.00903 [astro-ph.HE] .
  * Bavera  _et al._ [2020] S. S. Bavera, T. Fragos, Y. Qin, E. Zapartas, C. J. Neijssel, I. Mandel, A. Batta, S. M. Gaebel, C. Kimball, and S. Stevenson, The origin of spin in binary black holes: Predicting the distributions of the main observables of Advanced LIGO, Astron. Astrophys. 635, A97 (2020), arXiv:1906.12257 [astro-ph.HE] .
  * Broekgaarden  _et al._ [2019] F. S. Broekgaarden, S. Justham, S. E. de Mink, J. Gair, I. Mandel, S. Stevenson, J. W. Barrett, A. Vigna-Gómez, and C. J. Neijssel, STROOPWAFEL: Simulating rare outcomes from astrophysical populations, with application to gravitational-wave sources, Mon. Not. Roy. Astron. Soc. 490, 5228 (2019), arXiv:1905.00910 [astro-ph.HE] .
  * Portegies Zwart and McMillan [2000] S. F. Portegies Zwart and S. L. W. McMillan, Black Hole Mergers in the Universe, ApJ 528, L17 (2000), arXiv:astro-ph/9910061 [astro-ph] .
  * Antonini and Gieles [2020] F. Antonini and M. Gieles, Merger rate of black hole binaries from globular clusters: theoretical error bars and comparison to gravitational wave data, arXiv e-print (2020), arXiv:2009.01861 [astro-ph.HE] .
  * Santoliquido  _et al._ [2020] F. Santoliquido, M. Mapelli, N. Giacobbo, Y. Bouffanais, and M. C. Artale, The cosmic merger rate density of compact objects: impact of star formation, metallicity, initial mass function and binary evolution, arXiv e-print (2020), arXiv:2009.03911 [astro-ph.HE] .
  * Rodriguez  _et al._ [2015] C. L. Rodriguez, M. Morscher, B. Pattabiraman, S. Chatterjee, C.-J. Haster, and F. A. Rasio, Binary Black Hole Mergers from Globular Clusters: Implications for Advanced LIGO, Phys. Rev. Lett. 115, 051101 (2015), [Erratum: Phys.Rev.Lett. 116, 029901 (2016)], arXiv:1505.00792 [astro-ph.HE] .
  * Rodriguez  _et al._ [2016] C. L. Rodriguez, S. Chatterjee, and F. A. Rasio, Binary Black Hole Mergers from Globular Clusters: Masses, Merger Rates, and the Impact of Stellar Evolution, Phys. Rev. D 93, 084029 (2016), arXiv:1602.02444 [astro-ph.HE] .
  * Rodriguez and Loeb [2018] C. L. Rodriguez and A. Loeb, Redshift Evolution of the Black Hole Merger Rate from Globular Clusters, Astrophys. J. Lett. 866, L5 (2018), arXiv:1809.01152 [astro-ph.HE] .
  * Di Carlo  _et al._ [2019] U. N. Di Carlo, N. Giacobbo, M. Mapelli, M. Pasquato, M. Spera, L. Wang, and F. Haardt, Merging black holes in young star clusters, Mon. Not. Roy. Astron. Soc. 487, 2947 (2019), arXiv:1901.00863 [astro-ph.HE] .
  * Kremer  _et al._ [2020] K. Kremer, M. Spera, D. Becker, S. Chatterjee, U. N. Di Carlo, G. Fragione, C. L. Rodriguez, C. S. Ye, and F. A. Rasio, Populating the upper black hole mass gap through stellar collisions in young star clusters, Astrophys. J. 903, 45 (2020), arXiv:2006.10771 [astro-ph.HE] .
  * Bartos  _et al._ [2017] I. Bartos, B. Kocsis, Z. Haiman, and S. Márka, Rapid and Bright Stellar-mass Binary Black Hole Mergers in Active Galactic Nuclei, Astrophys. J. 835, 165 (2017), arXiv:1602.03831 [astro-ph.HE] .
  * Yi and Cheng [2019] S.-X. Yi and K. Cheng, Where Are the Electromagnetic-wave Counterparts of Stellar-mass Binary Black Hole Mergers?, Astrophys. J. Lett. 884, L12 (2019), arXiv:1909.08384 [astro-ph.HE] .
  * Yang  _et al._ [2019] Y. Yang _et al._ , Hierarchical Black Hole Mergers in Active Galactic Nuclei, Phys. Rev. Lett. 123, 181101 (2019), arXiv:1906.09281 [astro-ph.HE] .
  * Yang  _et al._ [2020] Y. Yang, I. Bartos, Z. Haiman, B. Kocsis, S. Márka, and H. Tagawa, Cosmic Evolution of Stellar-mass Black Hole Merger Rate in Active Galactic Nuclei, Astrophys. J. 896, 138 (2020), arXiv:2003.08564 [astro-ph.HE] .
  * Gröbner  _et al._ [2020] M. Gröbner, W. Ishibashi, S. Tiwari, M. Haney, and P. Jetzer, Binary black hole mergers in AGN accretion discs: gravitational wave rate density estimates, Astron. Astrophys. 638, A119 (2020), arXiv:2005.03571 [astro-ph.GA] .
  * Tagawa  _et al._ [2020a] H. Tagawa, Z. Haiman, and B. Kocsis, Formation and Evolution of Compact Object Binaries in AGN Disks, Astrophys. J. 898, 25 (2020a), arXiv:1912.08218 [astro-ph.GA] .
  * Tagawa  _et al._ [2020b] H. Tagawa, B. Kocsis, Z. Haiman, I. Bartos, K. Omukai, and J. Samsing, Mass-gap Mergers in Active Galactic Nuclei, arXiv e-print (2020b), arXiv:2012.00011 [astro-ph.HE] .
  * Tagawa  _et al._ [2020c] H. Tagawa, Z. Haiman, I. Bartos, and B. Kocsis, Spin Evolution of Stellar-mass Black Hole Binaries in Active Galactic Nuclei, Astrophys. J. 899, 26 (2020c), arXiv:2004.11914 [astro-ph.HE] .
  * Samsing  _et al._ [2020] J. Samsing, I. Bartos, D. D’Orazio, Z. Haiman, B. Kocsis, N. Leigh, B. Liu, M. Pessah, and H. Tagawa, Active Galactic Nuclei as Factories for Eccentric Black Hole Mergers, Arxiv e-print (2020), arXiv:2010.09765 [astro-ph.HE] .
  * Kinugawa  _et al._ [2014] T. Kinugawa, K. Inayoshi, K. Hotokezaka, D. Nakauchi, and T. Nakamura, Possible Indirect Confirmation of the Existence of Pop III Massive Stars by Gravitational Wave, Mon. Not. Roy. Astron. Soc. 442, 2963 (2014), arXiv:1402.6672 [astro-ph.HE] .
  * Kinugawa  _et al._ [2016] T. Kinugawa, A. Miyamoto, N. Kanda, and T. Nakamura, The detection rate of inspiral and quasi-normal modes of Population III binary black holes which can confirm or refute the general relativity in the strong gravity region, Mon. Not. Roy. Astron. Soc. 456, 1093 (2016), arXiv:1505.06962 [astro-ph.SR] .
  * Hartwig  _et al._ [2016] T. Hartwig, M. Volonteri, V. Bromm, R. S. Klessen, E. Barausse, M. Magg, and A. Stacy, Gravitational Waves from the Remnants of the First Stars, Mon. Not. Roy. Astron. Soc. 460, L74 (2016), arXiv:1603.05655 [astro-ph.GA] .
  * Belczynski  _et al._ [2017] K. Belczynski, T. Ryu, R. Perna, E. Berti, T. Tanaka, and T. Bulik, On the likelihood of detecting gravitational waves from Population III compact object binaries, Mon. Not. Roy. Astron. Soc. 471, 4702 (2017), arXiv:1612.01524 [astro-ph.HE] .
  * De Luca  _et al._ [2021a] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, The Minimum Testable Abundance of Primordial Black Holes at Future Gravitational-Wave Detectors, ArXiv (2021a), arXiv:2106.13769 [astro-ph.CO] .
  * De Luca  _et al._ [2021b] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Bayesian Evidence for Both Astrophysical and Primordial Black Holes: Mapping the GWTC-2 Catalog to Third-Generation Detectors, JCAP 05, 003, arXiv:2102.03809 [astro-ph.CO] .
  * Abbott  _et al._ [2017] B. P. Abbott _et al._ (LIGO Scientific), Exploring the Sensitivity of Next Generation Gravitational Wave Detectors, Class. Quant. Grav. 34, 044001 (2017), arXiv:1607.08697 [astro-ph.IM] .
  * Reitze  _et al._ [2019] D. Reitze _et al._ , Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833 [astro-ph.IM] .
  * Evans  _et al._ [2021] M. Evans _et al._ , A horizon study for cosmic explorer science, observatories, and community, dcc.cosmicexplorer.org/CE-P2100003/public (2021).
  * Punturo  _et al._ [2010] M. Punturo _et al._ , The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27, 194002 (2010).
  * Maggiore  _et al._ [2020] M. Maggiore _et al._ , Science Case for the Einstein Telescope, JCAP 03, 050, arXiv:1912.02622 [astro-ph.CO] .
  * Mirbabayi  _et al._ [2020] M. Mirbabayi, A. Gruzinov, and J. Noreña, Spin of Primordial Black Holes, JCAP 03, 017, arXiv:1901.05963 [astro-ph.CO] .
  * De Luca  _et al._ [2019] V. De Luca, V. Desjacques, G. Franciolini, A. Malhotra, and A. Riotto, The initial spin probability distribution of primordial black holes, JCAP 05, 018, arXiv:1903.01179 [astro-ph.CO] .
  * Bianchi  _et al._ [2018] E. Bianchi, A. Gupta, H. M. Haggard, and B. S. Sathyaprakash, Small Spins of Primordial Black Holes from Random Geometries: Bekenstein-Hawking Entropy and Gravitational Wave Observations, arXiv (2018), arXiv:1812.05127 [gr-qc] .
  * Note [1] Throughout this study, we use a $\Lambda$CDM cosmology based on the Planck 2018 results [106] to convert luminosity distance into redshift.
  * Skilling [2006] J. Skilling, Nested sampling for general Bayesian computation, Bayesian Analysis 1, 833 (2006).
  * Speagle [2020] J. S. Speagle, DYNESTY: a dynamic nested sampling package for estimating Bayesian posteriors and evidences, MNRAS 493, 3132 (2020), arXiv:1904.02180 [astro-ph.IM] .
  * Ashton  _et al._ [2019] G. Ashton _et al._ , BILBY: A user-friendly Bayesian inference library for gravitational-wave astronomy, Astrophys. J. Suppl. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM] .
  * Vallisneri [2008] M. Vallisneri, Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects, Phys. Rev. D 77, 042001 (2008), arXiv:gr-qc/0703086 .
  * Rodriguez  _et al._ [2014] C. L. Rodriguez, B. Farr, V. Raymond, W. M. Farr, T. B. Littenberg, D. Fazi, and V. Kalogera, Basic Parameter Estimation of Binary Neutron Star Systems by the Advanced LIGO/Virgo Network, Astrophys. J. 784, 119 (2014), arXiv:1309.3273 [astro-ph.HE] .
  * Pratten  _et al._ [2021] G. Pratten _et al._ , Computationally efficient models for the dominant and subdominant harmonic modes of precessing binary black holes, Phys. Rev. D 103, 104056 (2021), arXiv:2004.06503 [gr-qc] .
  * Usman  _et al._ [2019] S. A. Usman, J. C. Mills, and S. Fairhurst, Constraining the Inclinations of Binary Mergers from Gravitational-wave Observations, Astrophys. J. 877, 82 (2019), arXiv:1809.10727 [gr-qc] .
  * Chen  _et al._ [2019] H.-Y. Chen, S. Vitale, and R. Narayan, Viewing angle of binary neutron star mergers, Phys. Rev. X 9, 031028 (2019), arXiv:1807.05226 [astro-ph.HE] .
  * Bromm [2006] V. Bromm, High-redshift gamma-ray bursts from population III progenitors, Astrophys. J. 642, 382 (2006), arXiv:astro-ph/0509303 .
  * de Souza  _et al._ [2011] R. S. de Souza, N. Yoshida, and K. Ioka, Population III.1 and III.2 Gamma-Ray Bursts: Constraints on the Event Rate for Future Radio and X-ray Surveys, Astron. Astrophys. 533, A32 (2011), arXiv:1105.2395 [astro-ph.CO] .
  * Koushiappas and Loeb [2017] S. M. Koushiappas and A. Loeb, Maximum redshift of gravitational wave merger events, Phys. Rev. Lett. 119, 221104 (2017), arXiv:1708.07380 [astro-ph.CO] .
  * Mocz  _et al._ [2020] P. Mocz _et al._ , Galaxy formation with BECDM – II. Cosmic filaments and first galaxies, Mon. Not. Roy. Astron. Soc. 494, 2027 (2020), arXiv:1911.05746 [astro-ph.CO] .
  * Note [2] Although, we note that there are studies suggesting an earlier $(z\gtrsim 50)$ [107] or a later $(z\lesssim 20)$ [108] formation of Pop III stars.
  * Inayoshi  _et al._ [2017] K. Inayoshi, R. Hirai, T. Kinugawa, and K. Hotokezaka, Formation pathway of Population III coalescing binary black holes through stable mass transfer, Mon. Not. Roy. Astron. Soc. 468, 5020 (2017), arXiv:1701.04823 [astro-ph.HE] .
  * Liu and Bromm [2020a] B. Liu and V. Bromm, The Population III origin of GW190521, Astrophys. J. Lett. 903, L40 (2020a), arXiv:2009.11447 [astro-ph.GA] .
  * Liu and Bromm [2020b] B. Liu and V. Bromm, Gravitational waves from Population III binary black holes formed by dynamical capture, Mon. Not. Roy. Astron. Soc. 495, 2475 (2020b), arXiv:2003.00065 [astro-ph.CO] .
  * Kinugawa  _et al._ [2020] T. Kinugawa, T. Nakamura, and H. Nakano, Chirp Mass and Spin of Binary Black Holes from First Star Remnants, Mon. Not. Roy. Astron. Soc. 498, 3946 (2020), arXiv:2005.09795 [astro-ph.HE] .
  * Tanikawa  _et al._ [2021] A. Tanikawa, H. Susa, T. Yoshida, A. A. Trani, and T. Kinugawa, Merger rate density of Population III binary black holes below, above, and in the pair-instability mass gap, Astrophys. J. 910, 30 (2021), arXiv:2008.01890 [astro-ph.HE] .
  * Ng  _et al._ [2021] K. K. Y. Ng, S. Vitale, W. M. Farr, and C. L. Rodriguez, Probing multiple populations of compact binaries with third-generation gravitational-wave detectors, Astrophys. J. Lett. 913, L5 (2021), arXiv:2012.09876 [astro-ph.CO] .
  * Vitale  _et al._ [2017] S. Vitale, D. Gerosa, C.-J. Haster, K. Chatziioannou, and A. Zimmerman, Impact of Bayesian Priors on the Characterization of Binary Black Hole Coalescences, Phys. Rev. Lett. 119, 251103 (2017), arXiv:1707.04637 [gr-qc] .
  * Zevin  _et al._ [2020] M. Zevin, C. P. L. Berry, S. Coughlin, K. Chatziioannou, and S. Vitale, You Can’t Always Get What You Want: The Impact of Prior Assumptions on Interpreting GW190412, Astrophys. J. Lett. 899, L17 (2020), arXiv:2006.11293 [astro-ph.HE] .
  * Bhagwat  _et al._ [2021] S. Bhagwat, V. De Luca, G. Franciolini, P. Pani, and A. Riotto, The importance of priors on LIGO-Virgo parameter estimation: the case of primordial black holes, JCAP 01, 037, arXiv:2008.12320 [astro-ph.CO] .
  * Vikaeus  _et al._ [2021] A. Vikaeus, E. Zackrisson, D. Schaerer, E. Visbal, E. Fransson, S. Malhotra, J. Rhoads, and M. Sahlén, Conditions for detecting lensed Population III galaxies in blind surveys with the James Webb Space Telescope, the Roman Space Telescope and Euclid, arXiv e-prints , arXiv:2107.01230 (2021), arXiv:2107.01230 [astro-ph.GA] .
  * Sun  _et al._ [2021] G. Sun, J. Mirocha, R. H. Mebane, and S. R. Furlanetto, Revealing the formation histories of the first stars with the cosmic near-infrared background, arXiv e-prints , arXiv:2107.09324 (2021), arXiv:2107.09324 [astro-ph.GA] .
  * Tanvir  _et al._ [2021] N. R. Tanvir, E. Le Floc’h, L. Christensen, J. Caruana, R. Salvaterra, G. Ghirlanda, B. Ciardi, U. Maio, V. D’Odorico, E. Piedipalumbo, S. Campana, P. Noterdaeme, L. Graziani, L. Amati, Z. Bagoly, L. G. Balázs, S. Basa, E. Behar, E. Bozzo, A. De Cia, M. Della Valle, M. De Pasquale, F. Frontera, A. Gomboc, D. Götz, I. Horvath, R. Hudec, S. Mereghetti, P. T. O’Brien, J. P. Osborne, S. Paltani, P. Rosati, O. Sergijenko, E. R. Stanway, D. Szécsi, L. V. Toth, Y. Urata, S. Vergani, and S. Zane, Exploration of the high-redshift universe enabled by THESEUS, arXiv e-prints , arXiv:2104.09532 (2021), arXiv:2104.09532 [astro-ph.IM] .
  * Mukherjee  _et al._ [2021] S. Mukherjee, M. S. P. Meinema, and J. Silk, Prospects of discovering sub-solar primordial black holes using the stochastic gravitational wave background from third-generation detectors, arXiv (2021), arXiv:2107.02181 [astro-ph.CO] .
  * Farr  _et al._ [2015] W. M. Farr, J. R. Gair, I. Mandel, and C. Cutler, Counting And Confusion: Bayesian Rate Estimation With Multiple Populations, Phys. Rev. D 91, 023005 (2015), arXiv:1302.5341 [astro-ph.IM] .
  * Wysocki  _et al._ [2019] D. Wysocki, J. Lange, and R. O’Shaughnessy, Reconstructing phenomenological distributions of compact binaries via gravitational wave observations, Phys. Rev. D 100, 043012 (2019), arXiv:1805.06442 [gr-qc] .
  * Thrane and Talbot [2019] E. Thrane and C. Talbot, An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models, Publ. Astron. Soc. Austral. 36, e010 (2019), [Erratum: Publ.Astron.Soc.Austral. 37, e036 (2020)], arXiv:1809.02293 [astro-ph.IM] .
  * Mandel  _et al._ [2019] I. Mandel, W. M. Farr, and J. R. Gair, Extracting distribution parameters from multiple uncertain observations with selection biases, Mon. Not. Roy. Astron. Soc. 486, 1086 (2019), arXiv:1809.02063 [physics.data-an] .
  * Vitale [2020] S. Vitale, One, No One, and One Hundred Thousand – Inferring the properties of a population in presence of selection effects, arXiv e-print (2020), arXiv:2007.05579 [astro-ph.IM] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO] .
  * Trenti and Stiavelli [2009] M. Trenti and M. Stiavelli, The Formation Rates of Population III Stars and Chemical Enrichment of Halos during the Reionization Era, Astrophys. J. 694, 879 (2009), arXiv:0901.0711 [astro-ph.CO] .
  * Tornatore  _et al._ [2007] L. Tornatore, S. Borgani, K. Dolag, and F. Matteucci, Chemical enrichment of galaxy clusters from hydrodynamical simulations, Mon. Not. Roy. Astron. Soc. 382, 1050 (2007), arXiv:0705.1921 [astro-ph] .

Supplementary Materials – In Fig. 4, we show $p_{\rm
tot}(z|f^{\mathrm{PBH}}_{\mathrm{III}})$ for 5 different values of
$f^{\mathrm{PBH}}_{\mathrm{III}}$. The appearance of the discontinuity at
$z_{\mathrm{crit}}$ is caused by the piecewise nature of the Pop III BBH
merger rate density in Eq. (3) of the Letter.

Given that the peak of the merger rate for BBHs from Pop III stars is at
$z\sim 12$ (Fig. 4), one might explore values of $z_{\mathrm{crit}}$ lower
than what we used in the body of the Letter. Below, we report a version of
Table I obtained with a less conservative $z_{\mathrm{crit}}=20$. This result
in a larger number of sources that clear the $P_{p}>0.9$ criterion, and even
yields sources for which all of the posterior is above $z_{\mathrm{crit}}$.

Figure 4: Prior based on the mixture model $p_{\rm tot}$ (solid lines), ranging from $f^{\mathrm{PBH}}_{\mathrm{III}}=100$ (blue), 10 (orange), 1 (green), 0.1 (red), and 0.01 (purple). The dashed lines and dashed-dotted lines represent the individual contribution from PBH mergers and Pop III BBH mergers (which is truncated at $z_{\mathrm{crit}}$), respectively.  Table 3: Same as Table I but with $z_{\mathrm{crit}}=20$ $M_{\rm tot}$ | $q$ | $\iota$ | $z$ | $\rho_{\rm CE}$ | $\rho_{\rm CES}$ | $\rho_{\rm ET}$ | $\rho_{\rm net}$ | $P_{p}$
---|---|---|---|---|---|---|---|---
10 | 1 | 0 | 40 | 12.1 | 5.8 | 3.7 | 13.9 | 0.92
10 | 1 | 0 | 50 | 11.3 | 5.4 | 3.4 | 13.0 | 0.98
20 | 1 | 0 | 30 | 22.7 | 10.6 | 7.0 | 26.0 | 0.96
20 | 1 | 0 | 40 | 19.9 | 8.5 | 6.0 | 22.5 | 1.00
20 | 1 | 0 | 50 | 16.6 | 7.2 | 5.9 | 19.1 | 0.99
20 | 2 | 0 | 30 | 19.4 | 9.0 | 6.2 | 22.3 | 0.99
20 | 2 | 0 | 40 | 16.5 | 7.3 | 5.5 | 18.9 | 0.99
20 | 2 | 0 | 50 | 13.6 | 5.8 | 5.2 | 15.7 | 0.99
20 | 3 | 0 | 30 | 15.7 | 7.2 | 5.1 | 18.0 | 0.98
20 | 3 | 0 | 40 | 13.0 | 5.7 | 4.5 | 14.9 | 1.00
20 | 4 | 0 | 30 | 13.0 | 5.8 | 4.4 | 14.9 | 0.98
20 | 5 | 0 | 30 | 11.0 | 4.9 | 3.8 | 12.6 | 0.99
20 | 1 | $\pi/6$ | 30 | 19.7 | 8.7 | 5.7 | 22.3 | 0.96
20 | 1 | $\pi/6$ | 40 | 17.0 | 7.3 | 5.1 | 19.2 | 0.99
20 | 1 | $\pi/6$ | 50 | 14.0 | 6.3 | 5.2 | 16.2 | 1.00
20 | 2 | $\pi/6$ | 30 | 17.3 | 7.6 | 5.0 | 19.6 | 0.95
20 | 2 | $\pi/6$ | 40 | 14.2 | 6.1 | 4.4 | 16.1 | 0.99
20 | 2 | $\pi/6$ | 50 | 11.6 | 5.1 | 4.5 | 13.5 | 0.99
20 | 3 | $\pi/6$ | 30 | 14.3 | 6.2 | 4.3 | 16.1 | 0.97
20 | 3 | $\pi/6$ | 40 | 11.1 | 5.0 | 4.0 | 12.8 | 0.99
20 | 4 | $\pi/6$ | 30 | 11.7 | 5.5 | 4.1 | 13.5 | 0.97
20 | 1 | $\pi/3$ | 30 | 13.1 | 5.7 | 3.3 | 14.7 | 0.98
20 | 1 | $\pi/3$ | 40 | 11.2 | 4.7 | 3.0 | 12.5 | 1.00
20 | 2 | $\pi/3$ | 30 | 12.2 | 5.4 | 3.0 | 13.7 | 0.95
40 | 1 | 0 | 30 | 27.6 | 12.6 | 13.3 | 33.1 | 0.98
40 | 1 | 0 | 40 | 8.8 | 8.2 | 19.8 | 23.2 | 0.99
40 | 1 | 0 | 50 | 3.0 | 4.2 | 17.8 | 18.5 | 0.99
40 | 2 | 0 | 30 | 21.5 | 10.1 | 12.2 | 26.8 | 0.98
40 | 2 | 0 | 40 | 5.6 | 6.0 | 17.2 | 19.0 | 0.98
40 | 2 | 0 | 50 | 2.1 | 3.0 | 14.5 | 15.0 | 0.98
40 | 3 | 0 | 30 | 14.6 | 7.8 | 11.5 | 20.2 | 0.96
40 | 3 | 0 | 40 | 3.2 | 4.0 | 13.7 | 14.7 | 0.98
40 | 4 | 0 | 30 | 10.1 | 6.2 | 10.7 | 16.0 | 0.97
40 | 5 | 0 | 30 | 7.6 | 5.0 | 9.5 | 13.2 | 0.99
40 | 1 | $\pi/6$ | 30 | 23.0 | 10.8 | 12.0 | 28.1 | 0.98
40 | 1 | $\pi/6$ | 40 | 7.4 | 6.9 | 16.8 | 19.6 | 0.99
40 | 1 | $\pi/6$ | 50 | 2.6 | 3.5 | 15.1 | 15.7 | 0.99
40 | 2 | $\pi/6$ | 30 | 19.2 | 9.0 | 10.5 | 23.7 | 0.93
40 | 2 | $\pi/6$ | 40 | 5.2 | 5.6 | 14.9 | 16.8 | 0.96
40 | 2 | $\pi/6$ | 50 | 2.2 | 3.0 | 12.8 | 13.3 | 0.98
40 | 3 | $\pi/6$ | 40 | 3.4 | 4.2 | 12.2 | 13.4 | 0.96
40 | 4 | $\pi/6$ | 30 | 10.2 | 6.3 | 9.4 | 15.2 | 0.92
40 | 1 | $\pi/3$ | 30 | 15.3 | 6.9 | 6.6 | 18.0 | 1.00
40 | 1 | $\pi/3$ | 40 | 6.0 | 4.6 | 9.5 | 12.1 | 1.00
40 | 2 | $\pi/3$ | 30 | 13.8 | 6.3 | 5.9 | 16.3 | 0.97
40 | 1 | $\pi/2$ | 30 | 11.5 | 4.6 | 2.8 | 12.7 | 1.00
80 | 1 | 0 | 30 | 2.7 | 4.5 | 29.1 | 29.5 | 0.98
80 | 1 | 0 | 40 | 0.8 | 1.4 | 14.1 | 14.2 | 0.93
80 | 2 | 0 | 30 | 1.9 | 3.2 | 22.4 | 22.7 | 0.97
80 | 3 | 0 | 30 | 1.2 | 2.0 | 15.7 | 15.9 | 0.94
80 | 1 | $\pi/6$ | 30 | 2.3 | 3.8 | 24.6 | 25.0 | 0.99
80 | 1 | $\pi/3$ | 30 | 1.3 | 2.7 | 14.8 | 15.1 | 1.00
80 | 2 | $\pi/3$ | 30 | 1.4 | 3.0 | 13.4 | 13.8 | 0.97