Constraining the Merger History of Primordial-Black-Hole Binaries from GWTC-3 Lang Liu1 2Zhi-Qiang You1 2yYou Wu3zand Zu-Cheng Chen1 2x

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Constraining the Merger History of Primordial-Black-Hole Binaries from GWTC-3

Lang Liu,1, 2, ∗Zhi-Qiang You,1, 2, †You Wu,3, ‡and Zu-Cheng Chen1, 2, §

1Department of Astronomy, Beijing Normal University, Beijing 100875, China

2Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai 519087, China

3College of Mathematics and Physics, Hunan University of Arts and Science, Changde, 415000, China

Primordial black holes (PBHs) can be not only cold dark matter candidates but also progenitors

of binary black holes observed by LIGO-Virgo-KAGRA (LVK) Collaboration. The PBH mass can

be shifted to the heavy distribution if multi-merger processes occur. In this work, we constrain the

merger history of PBH binaries using the gravitational wave events from the third Gravitational-

Wave Transient Catalog (GWTC-3). Considering four commonly used PBH mass functions, namely

the log-normal, power-law, broken power-law, and critical collapse forms, we ﬁnd that the multi-

merger processes make a subdominant contribution to the total merger rate. Therefore, the eﬀect

of merger history can be safely ignored when estimating the merger rate of PBH binaries. We also

ﬁnd that GWTC-3 is best ﬁtted by the log-normal form among the four PBH mass functions and

conﬁrm that the stellar-mass PBHs cannot dominate cold dark matter.

I. INTRODUCTION

The successful detection of gravitational waves (GWs)

from compact binary coalescences [1–3] has led us into

a new era of GW astronomy. According to the recently

released third GW Transient Catalog (GWTC-3) [3] by

LIGO-Virgo-KAGRA (LVK) Collaboration, there are 90

GW events detected during the ﬁrst three observing runs.

Most of these events are categorized as binary black

hole (BBH) mergers, and the BBHs detected by LVK

have a broad mass distribution. The heaviest event,

GW190521 [4], has component masses m1= 85+21

−14M

and m2= 66+17

−18M. Both masses lie within upper black

hole mass gap originated from pulsation pair-instability

supernovae [5], and current modelling places the lower

cutoﬀ of the mass gap at ∼50 ±4M[5–9]. Even ac-

counting for the statistical uncertainties, it still implies

at least m1is well within the mass gap and cannot orig-

inate directly from a stellar progenitor [10]. Therefore,

the heavy event GW190521 greatly challenges the stellar

evolution scenario of astrophysical black holes.

Besides the astrophysical black holes, another possible

explanation for the LVK BBHs is the primordial black

holes (PBHs) [11–15]. PBHs are black holes formed in

the very early Universe through the gravitational collapse

of the primordial density ﬂuctuations [16,17]. Recently,

PBHs have attracted considerable attention [18–33] be-

cause they can be not only the sources of LVK detec-

tions [11,12], but also candidates of cold dark matter

(CDM) [34] and the seeds for galaxy formation [35,36].

The formation of PBHs would inevitably accompany the

production of scalar-induced GWs [37–45]. Recent stud-

ies [15,46] show that the BBHs from GWTC-3 are con-

sistent with the PBH scenario, and the abundance of

∗liulang@bnu.edu.cn

†Corresponding author: zhiqiang.you@bnu.edu.cn

‡youwuphy@gmail.com

§Corresponding author: zucheng.chen@bnu.edu.cn

PBH in CDM, fpbh, should be in the order of O(10−3)

to explain LVK BBHs. In particular, the merger rate

for GW190521 derived from the PBH model is consistent

with that inferred by LVK, indicating that GW190521

can be a PBH binary [15,24].

Accurately estimating the merger rate distribution of

PBH binaries can be crucial to extract the PBH popu-

lation parameters from GW data. Ref. [21] studies the

multi-merger processes of PBH binaries and show that

the merger history of PBH binaries may shift the mass

distribution from light mass to heavy mass depending

on the values of population parameters. Ref. [47] then

infers the population parameters of PBH binaries by ac-

counting for the merger history eﬀect using 10 BBHs from

GWTC-1, ﬁnding that the eﬀect of merger history can be

safely ignored when estimating the merger rate of PBH

binaries. In this work, we use the LVK recent released

GWTC-3 data to constrain the eﬀect of merger history

on the merger rate of PBH binaries assuming all LVK

BBHs are of primordial origin. We extend the analyses

of Ref. [47] in several aspects. Firstly, we use a puriﬁed

subset of GWTC-3, which expands GWTC-1 with al-

most six times more BBH events. The GWTC-3 events

expand the mass and redshift coverage and can allevi-

ate the statistical bias by including signiﬁcantly more

BBHs. Secondly, Ref. [47] only considers the PBH mass

functions with the log-normal and power-law forms. We

do more comprehensive analyses by including the broken

power-law and critical collapse PBH mass functions that

were not considered in Ref. [47]. It is claimed by Ref. [48]

that a broken power-law can ﬁt the GW data better than

the log-normal form. Lastly, we consider the redshift dis-

tribution of the merger rate that is ignored in Ref. [47].

The aforementioned reasons have inspired us to explore

the possibility that the heavy black holes detected by

LVK have been formed, at least in part, through second-

generation mergers. This is because the second-merger

process has the potential to increase the mass distribu-

tion to a higher value. A precise assessment of the inﬂu-

ence of second-generation mergers on mass distribution

demands a meticulous analysis of the data, as has been

arXiv:2210.16094v2 [astro-ph.CO] 2 Apr 2023

conducted in this study.

We organize the rest paper as follows. In Sec. II, we

brieﬂy review the calculation of the merger rate of PBH

binaries by accounting for the merger history eﬀect. In

Sec. III, we describe the hierarchical Bayesian framework

used to infer the PBH population parameters from GW

data. In Sec. IV, we consider four commonly used PBH

mass functions and present the results. Finally, we give

conclusions in Sec. V.

II. MERGER RATE DENSITY DISTRIBUTION

OF PBH BINARIES

In this section, we will outline the calculation of merger

rate density when considering the PBH merger history

eﬀect. We refer to Ref. [21] for more details.

The BBHs observed by LVK suggest that BHs should

have a broad mass distribution, so we consider an ex-

tended mass function for PBHs. Here, we demand the

probability distribution function of PBH mass, P(m), be

normalized such that

Z∞

P(m) dm= 1.(1)

Assuming the fraction of PBHs in CDM is fpbh, we can

estimate the abundance of PBHs in the mass interval

(m, m + dm) as [49]

0.85fpbh P(m) dm. (2)

The coeﬃcient 0.85 is roughly the fraction of CDM in

the non-relativistic matter, including both CDM and

baryons. Following Ref. [21], we may deﬁne an average

PBH mass, mpbh, as

mpbh

=ZP(m)

mdm. (3)

Then, we can obtain the average number density of PBHs

with mass min the total number density of PBHs, F(m),

by [21]

F(m) = P(m)mpbh

m.(4)

We can now estimate the merger rate densities of PBH

binaries by considering the merger history eﬀect. We as-

sume that PBHs are randomly distributed following a

spatial Poisson distribution in the early Universe when

they decouple from the cosmic background evolution

[12,50,51]. The two nearest PBHs would attract each

other because of the gravitational interactions. These

two PBHs would obtain the angular momentum from the

torque of other PBHs and form a PBH binary after de-

coupling from the cosmic expansion. The binary would

emit gravitational radiations and eventually merge.

We do not intend to give a detailed derivation but

quote the results from Ref. [21] here. The merger rate

density from ﬁrst-merger process, R1(t, mi, mj), is given

by [21]

R1(t, mi, mj) = Zˆ

R1dml,(5)

where miand mjare the masses of the merging binary,

mlis the mass of the third black hole that is closest to

the merging binary, and

R1(t, mi, mj, ml)≡1.32 ×106×t

t0−34

37 fpbh

mpbh 53

×m−21

l(mimj)3

37 (mi+mj)36

37 F(mi)F(mj)F(ml).

(6)

Here, tis the cosmic time, and t0is the present cosmic

time. Similarly, the merger rate density from second-

merger process, R2(t, mi, mj), is given by [21]

R2(t, mi, mj) = 1

2Zˆ

R2(t, mi−me, me, mj, ml) dmldme

2Zˆ

R2(t, mj−me, me, mi, ml) dmldme,

(7)

where meis the mass of the fourth black hole that is

closest to the merging binary, and

R2(t, mi, mj, mk, ml)=1.59 ×104×t

t0−31

37 fpbh

mpbh 69

×m

km−42

l(mi+mj)6

37 (mi+mj+mk)72

×F(mi)F(mj)F(mk)F(ml).

(8)

We only consider the eﬀect of merger history up to the

second-merger process. We have veriﬁed that the frac-

tion of the third merger rate over the second merger rate

is less than 0.005. Therefore the total merger rate den-

sity, R(t, mi, mj), of PBH binaries at cosmic time twith

masses miand mjis

R(t, mi, mj) = X

n=1,2Rn(t, mi, mj),(9)

and the total merger rate is

R(t) = ZR(t, mi, mj)dmidmj=X

n=1,2

Rn(t),(10)

where

Rn(t) = ZRn(t, mi, mj)dmidmj.(11)

All the above-mentioned merger rate (density) is mea-

sured at the source frame. We should emphasize that

although R2(t) should be smaller than R1(t) as ex-

pected, R2(t, mi, mj) is not necessarily be smaller than

R1(t, mi, mj) [21].

Parameter Description Prior

fpbh Abundance of PBH in CDM log-U(−4,0)

Lognormal PBH mass function

McCentral mass in M.U(5,50)

σMass width. U(0.1,2)

Power-law PBH mass function

Mmin Lower mass cut-oﬀ in M.U(3,10)

αPower-law index. U(1.05,4)

Broken Power-law PBH mass function

m∗Peak mass in M.U(5,50)

α1First power-law index. U(0,3)

α2Second power-law index. U(1,10)

Critical collapse (CC) PBH mass function

MfHorizon mass scale in M.U(1,50)

αUniversal exponent. U(0,5)

TABLE I. Parameters and their prior distributions used in the

Bayesian parameter estimations. Here, Uand log-Udenote

uniform and log-uniform distributions, respectively.

III. HIERARCHICAL BAYESIAN INFERENCE

We adopt a hierarchical Bayesian approach to infer the

population parameters by marginalizing the uncertainty

in estimating individual event parameters. This section

describes the hierarchical Bayesian inference used in the

parameter estimations. The merger rate density (9) is

measured in the source frame, and we need to convert it

into the detector frame as

Rpop(θ|Λ) = 1

1 + z

dVc

dz R(θ|Λ),(12)

where zis the cosmological redshift, θ≡ {z, m1, m2},

Λ is a collection of fpbh and the parameters from mass

function P(m), and dVc/dz is the diﬀerential comoving

volume. The factor 1/(1 + z) converts time increments

from the source to the detector frame. We take the cos-

mological parameters from Planck 2018 [52].

Given the data, d={d1, d2,··· , dNobs }, of Nobs BBH

merger events, we model the total number of events as an

inhomogeneous Poisson process, yielding the likelihood

[53–55]

L(d|Λ) ∝NNobs

exp e−Nexp

Nobs

i=1 RL(di|θ)Rpop(θ|Λ)dθ

ξ(Λ) ,

(13)

where Nexp ≡Nexp(Λ) is the expected number of detec-

tions over the timespan of observation. Here L(di|θ) is

the individual event likelihood for the ith GW event that

can be derived from the individual event’s posterior by

reweighing with the prior on θ. Here, ξ(Λ) quantiﬁes se-

lection biases for a population with parameters Λ and is

deﬁned by

ξ(Λ) = ZPdet(θ)Rpop(θ|Λ) dθ, (14)

where Pdet(θ) is the detection probability that depends

on the source parameters θ. In practice, we use the sim-

ulated injections [56] to estimate ξ(Λ), and Eq. (14) can

be approximated by a Monte Carlo integral over found

injections [57]

ξ(Λ) ≈1

Ninj

Nfound

j=1

Rpop(θj|Λ)

pdraw(θj),(15)

where Ninj is the total number of injections, Nfound is the

number of successfully detected injections, and pdraw is

the probability density function from which the injections

are drawn. Using the posterior samples from each event,

we estimate the hyper-likelihood (13) as

L(d|Λ) ∝NNobs

exp e−Nexp

Nobs

i=1

ξ(Λ) Rpop(θ|Λ)

L(z),(16)

where h···i denotes the weighted average over posterior

samples of θ. The denominator d2

L(z) is the standard

priors used in the LVK analysis of individual events where

dLis the luminosity distance.

In this work, we incorporate the PBH population dis-

tribution (9) into the ICAROGW [58] package to estimate

the likelihood function (16), and use dynesty [59] sam-

pler called from Bilby [60,61] to sample over the pa-

rameter space. We use the GW events from GWTC-3 by

discarding events with false alarm rate larger than 1 yr−1

and events with the secondary component mass smaller

than 3Mto avoid contamination from putative events

involving neutron stars following Ref. [62]. A total of 69

GW events from GWTC-3 meet these criteria and the

posterior samples of these BBHs are publicly available

from Ref. [63].

IV. RESULTS

Based on the hierarchical statistical framework, we do

the parameter estimations for four diﬀerent PBH mass

functions commonly used in the literature. These mass

functions are the log-normal, power-law, broken power-

law, and critical collapse (CC) distributions, respectively.

We summarize the parameters and their prior distribu-

tions in Table I. Below we show the results for each of

the PBH mass functions.

A. Log-normal mass function

We ﬁrst consider a PBH mass function with the log-

normal form of [64]

P(m) = 1

√2πσm exp −ln2(m/Mc)

2σ2,(17)

where Mcrepresents the central mass of mP (m), and σ

characterizes the width of the mass spectrum. The log-

normal mass function can approximate a huge class of

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ConstrainingtheMergerHistoryofPrimordial-Black-HoleBinariesfromGWTC-3LangLiu,1,2,Zhi-QiangYou,1,2,yYouWu,3,zandZu-ChengChen1,2,x1DepartmentofAstronomy,BeijingNormalUniversity,Beijing100875,China2AdvancedInstituteofNaturalSciences,BeijingNormalUniversity,Zhuhai519087,China3CollegeofMathematicsandPhy...

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