On the Merger Rate of Primordial Black Holes in Cosmic Voids Saeed Fakhry1 2Seyed Sajad Tabasi3 2yand Javad T. Firouzjaee4 5 2z 1Department of Physics Shahid Beheshti University Evin Tehran 19839 Iran

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On the Merger Rate of Primordial Black Holes in Cosmic Voids

Saeed Fakhry,1, 2, ∗Seyed Sajad Tabasi,3, 2, †and Javad T. Firouzjaee4, 5, 2, ‡

1Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran

2PDAT Laboratory, Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

3Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran

4Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

5School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

(Dated: May 9, 2023)

Cosmic voids are known as underdense substructures of the cosmic web that cover a large volume

of the Universe. It is known that cosmic voids contain a small number of dark matter halos, so the

existence of primordial black holes (PBHs) in these secluded regions of the Universe is not unlikely. In

this work, we calculate the merger rate of PBHs in dark matter halos structured in cosmic voids and

determine their contribution to gravitational wave events resulting from black hole mergers recorded

by the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO)-Advanced Virgo

(aVirgo) detectors. Relying on the PBH scenario, the results of our analysis indicate that about

2∼3 annual events of binary black hole mergers out of all those recorded by the aLIGO-aVirgo

detectors should belong to cosmic voids. We also calculate the redshift evolution of the merger

rate of PBHs in cosmic voids. The results show that the evolution of the merger rate of PBHs has

minimum sensitivity to the redshift changes, which seems reasonable while considering the evolution

of cosmic voids. Finally, we specify the behavior of the merger rate of PBHs as a function of their

mass and fraction in cosmic voids and we estimate R(MPBH, fPBH) relation, which is well compatible

with our ﬁndings.

PACS numbers: 95.35.+d; 98.80.-k; 04.25.dg; 95.85.Sz

Keywords: Primordial Black Hole; Dark Matter; Cosmic Void; Merger Rate Per Halo.

I. INTRODUCTION

Primordial black holes (PBHs) are a special type of

black holes that form under unique conditions in the ear-

liest evolutionary stages of the Universe [1–3]. When pri-

mordial quantum ﬂuctuations exceed a threshold value,

they become qualiﬁed to directly collapse and lead to the

formation of PBHs. The formation of a black hole is itself

a fully general relativistic physical process and the evolu-

tion of non-linear density perturbations on superhorizon

scales and the threshold amplitude of curvature perturba-

tions on superhorizon scales for forming black holes were

investigated in several studies [4–10]. In addition, since

PBHs form in the early Universe, i.e., before the forma-

tion of stars and galaxies, and behave like cosmic ﬂuids

at large scales, they can be considered as potential candi-

dates for dark matter depending on their masses [11–17].

Moreover, they can provide seeds for the early formation

of supermassive black holes (see, e.g., [18–20]), and are

instrumental in the processes occurring in the early Uni-

verse, see e.g.,[21–23] for a review and references therein.

Theoretically, during the evolution of the Universe,

one can expect that PBHs could have been clustered in

dark matter halos. Also, due to their random spatial

distribution, it makes sense that PBHs in the dark mat-

ter halos are capable of encountering each other and/or

∗Electronic address: s fakhry@sbu.ac.ir

†Electronic address: sstabasi98@gmail.com

‡Electronic address: ﬁrouzjaee@kntu.ac.ir

other compact objects, forming binaries, emitting grav-

itational waves, and ﬁnally merging. After several de-

tections of gravitational waves associated with binary

black hole mergers by the Advanced Laser Interferome-

ter Gravitational-Wave Observatory (aLIGO)-Advanced

Virgo (aVirgo) detectors [24–26], the interest in PBHs

has increased recently. Most mergers are related to bi-

nary black holes with masses of (10 ∼30) M. In light

of this black hole mass range, which often exceeds the

astrophysics black hole mass spectrum, it has been sug-

gested that these binary black holes may have primordial

origins, see, e.g., [27–31].

To study the merger rate of PBHs in dark matter ha-

los, two main aspects should be considered. First, the

formation mechanism of PBHs, which is related to their

contribution to cold dark matter. Second, the physics of

dark matter halos within which the PBHs merge. Hence,

it can be expected that the concentration parameter of

dark matter halos can aﬀect the merger rate of PBHs by

modifying their velocity and density distributions. Also,

having a statistical view on large scales, it can be found

that the mass distribution of dark matter halos can also

have a special signiﬁcance in this case. In this regard,

it can be expected that various dark matter halo models

provide diﬀerent predictions of the merger rate of PBHs.

Having diﬀerent types of halo collapse models, the an-

alytically simple one is based on a spherical collapse,

which seems unable to provide accurate predictions of

local and statistical properties of dark matter halos in

some mass limits. To study the role of collapse con-

ditions, in our previous research, we have shown that

spherical-collapse dark matter halo models cannot ad-

arXiv:2210.13558v2 [astro-ph.CO] 3 Apr 2023

just the recorded black hole mergers during the third

observing run (O3) in the framework of the PBH sce-

nario, while more realistic halo models (e.g., those with

ellipsoidal-collapse) can generate consistent PBH merg-

ers with gravitational wave observations, see our previous

studies in Refs. [32–35] for more details.

On the other side, it is believed that the large-scale vis-

ible Universe appears to have a web-like structure called

the cosmic web. In some parts of the cosmic web, clus-

ters are separated by large, almost empty regions called

cosmic voids where the density of such regions is lower

than the average density of the Universe. Being a ma-

jor part of the cosmic web, cosmic voids are exception-

ally underdense regions containing matter, which evac-

uates them toward other regions. Researches in recent

years show that there are galaxies and dark matter halos

in cosmic voids [36–39]. According to some theoretical

models, dark matter particles are expected to emit de-

tectable gamma-ray signals as a result of their decay and

annihilation [40–44]. Additionally, a diﬀuse background

of gamma rays can be detected across the sky by the

present gamma-ray observatories [45]. This background

consists of unknown signals that remain after subtracting

the contribution from all possible astrophysical sources,

such as supermassive black holes and pulsars. Besides,

such signals have a non-uniform distribution at diﬀerent

spatial angles, which is adjustable to what is expected

from dark matter emission [46]. Accordingly, the way

of emitting signals related to the dark matter from over-

dense and underdense structures of the Universe has been

simulated [47]. The results indicate that although overall

dark matter signals from cosmic voids are weaker, they

are less contaminated by astrophysical sources, making

them easier to detect. In light of everything discussed

so far, the existence of PBHs in cosmic voids is not far

from expected. Therefore, it seems interesting to calcu-

late their merger rate in underdense environments with

minimal contamination by astrophysical sources.

In this work, we propose to calculate the merger rate

of PBHs in the medium of cosmic voids. In this respect,

the outline of the work is as follows. In Sec. II, we brieﬂy

discuss cosmic voids and the need to include them in

cosmological studies. Then, in Sec. III, we present a con-

venient dark matter halo model in cosmic voids and de-

scribe some related quantities like halo density proﬁle,

halo concentration parameter, and halo mass function.

Also, in Sec. IV, we calculate the merger rate of PBHs

in cosmic voids, as well as those in other structures. Fi-

nally, we discuss the results and summarize the ﬁndings

in Sec. V.

II. COSMIC VOIDS

The visible Universe on large scales seems to have a

web-like structure called the cosmic web. Such a struc-

ture is the result of the time evolution of primordial den-

sity ﬂuctuations. There are smaller structures inside the

cosmic web, including knots, ﬁlaments, sheets, and voids,

within which matter is distributed diﬀerently. Since the

initial density ﬁeld is a Gaussian random ﬁeld deﬁned

by the power spectrum of density ﬂuctuations, and since

density ﬂuctuations evolve due to gravity, the gravita-

tional ﬁeld increases the density contrast in the Universe.

As a result, parts of the Universe with stronger gravita-

tional ﬁelds become denser over time, while parts with

less density become even more empty [48].

In the light these arguments, most of the matter can

be distributed in knots, ﬁlaments, and sheets. However,

clusters are separated by large, almost empty regions

called cosmic voids. The density of such regions is lower

than the average density of the Univers. Actually, cos-

mic voids are underdense regions with a low density of

matter. According to recent cosmological studies, cosmic

voids can provide clues about cosmic mass distribution

and serve as a convenient medium for constraining cos-

mological parameters [49–51]. In addition, by taking ad-

vantage of the dynamics governing cosmic voids, many

studies have been conducted on baryon acoustic oscilla-

tions [52,53], dark matter-dark energy interaction [54],

cosmic microwave background [55], and many other hot

topics.

In Ref. [56], a comparison between the properties of

various galaxies within cosmic voids has been performed,

in which the number density of galaxies is less than 10%

of the mean density of the Universe. Also, to ﬁnd out the

properties of cosmic voids, several studies have been car-

ried out via N-body simulations [57–59]. By considering

the size, shape, and structure of cosmic voids, it can be

found that such regions might play a prominent role in

cosmological evolution because they cover a large volume

of the Universe.

Based on the cosmological perturbation theory in the

formation and evolution of large-scale structures, a vast

majority of matter budget is distributed in dark mat-

ter halos [60]. Numerical simulations and explanatory

studies indicate that the gravitational enhancement of

density ﬂuctuations leads to the formation of large-scale

structures [61]. Moreover, a few studies have mapped the

population of cosmic voids within the local Universe [62–

64]. Large regions of cosmic voids are related to the fore-

most noticeable viewpoint of the megaparsec-scale Uni-

verse. Cosmic voids are attributed to enormous regions

with sizes about (20 ∼50) Mpc h−1that have a lower

distribution of matter compared to other structures and

possess a signiﬁcant share of the Universe [65].

Although some studies have been performed on black

holes in cosmic voids, e.g., [66–68], no attention has been

paid to the merger rate PBHs and their evolution in such

regions. As mentioned earlier, it has been suggested that

there are galaxies and dark matter halos in cosmic voids.

Hence, it is likely that PBHs can be clustered in cos-

mic voids. Furthermore, research continues on develop-

ing new methods for ﬁnding galaxies, molecular gas, and

star formation in cosmic voids [69,70]. Cosmic voids

are interesting from a theoretical perspective because of

their huge volume and low density of matter [71]. In this

way, the behavior of dark matter and its candidates can

be studied more precisely by considering cosmic voids as

solitude regions free of possible astrophysical noises.

Moreover, we might face many challenges when inves-

tigating cosmic voids from the standpoints of observa-

tion, theory, and simulation. For instance, choosing the

right algorithm such as VoidFinder algorithm [72], ZOnes

Bordering On Voidness (ZOBOV) algorithm [73], and

DynamIcal Void Analysis (DIVA) algorithm [74] to ﬁnd

cosmic voids is very susceptible. On the other hand, the

existence of degeneracies between cosmological parame-

ters in cosmic void simulations cannot be ignored in any

way. Regarding this, some studies, e.g., [75], have tried to

resolve this issue. Furthermore, one must develop some

theoretical models that account for galaxy bias and its

impact on cosmic void characteristics. However, exten-

sive eﬀorts have been made in this regard [76–78].

The structure of cosmic voids is a secluded environ-

ment that makes them suitable for observational and

gravitational searches. Cosmic voids have less noise than

other structures. Thus, it provides a convenient oppor-

tunity for focusing on gravitational waves emitted from

such regions. Note that the sensitivity of gravitational

wave detectors is still not high enough to detect the ex-

act location of merger events. However, with the develop-

ment of instruments, this can be realized in the upcoming

future in such a way that one can witness the classiﬁca-

tion of merger events arising from the substructures of

the cosmic web.

III. HALO MODELS IN COSMIC VOIDS

Dark matter halos are nonlinear cosmological struc-

tures that spread in the Universe based on the dynam-

ics governing the formation and evolution of hierarchical

structures. They usually originate from physical condi-

tions under which the primordial density ﬂuctuations can

be qualiﬁed to separate from the expansion of the Uni-

verse and collapse due to the self-gravitational force [79].

In other words, the physical interpretation of the forma-

tion of cosmological structures can be deduced from a

dimensionless quantity called density contrast, which is

deﬁned as δ(r)≡[ρ(r)−¯ρ]/¯ρ. In this relation, ¯ρrep-

resents the mean density of the background, and ρ(r) is

the density of the overdense region at arbitrary point r.

According to this deﬁnition, which is derived from the ex-

cursion sets theory, density ﬂuctuations that exceed the

threshold value of overdensities, i.e., δc(z)'1.686(1+z),

can provide convenient conditions for the formation of

dark matter halos.

On the other hand, the size of cosmic voids and their

under-density nature turn them into suitable platforms

for investigating the primordial density ﬂuctuations and

the formation of cosmological structures such as dark

matter halos. Therefore, having proper knowledge about

the properties of dark matter halos that form in cosmic

voids is necessary [80]. In this regard, the cosmologi-

cal perturbation theory expresses the density distribution

of dark matter particles in galactic halos as a radius-

dependent function called the halo density proﬁle. To

justify the various ﬁts related to the data obtained from

the rotation curve of galaxies, many studies have been

carried out to provide a suitable density proﬁle [81–85].

One of the most popular density proﬁles was provided

by Navarro, Frenk, and White (NFW), which has the

following form [85]

ρ(r) = ρs

r/rs(1 + r/rs)2,(1)

where ρs=ρcritδcis the scaled density of the halo, ρcrit

is the critical density of the background Universe, and rs

is the scale radius of the halo. In addition to the NFW

proﬁle, the Einasto density proﬁle has been introduced

as [81]

ρ(r) = ρsexp −2

αr

rsα

−1,(2)

where αis the shape parameter. Although the mentioned

density proﬁles have been derived through diﬀerent meth-

ods, both of them are in good agreement with most of

the rotation curve data. In addition, another description

of the density proﬁle can be deduced from the concen-

tration parameter, which usually determines the central

density of dark matter halos and is deﬁned as

C≡rvir

,(3)

where rvir is the halo virial radius. In general, the halo

virial radius is attributed to a space that contains a vol-

ume whose density is 200 to 500 times the critical density

of the background Universe. Many attempts have been

performed to obtain a suitable concentration parameter

[86–89]. Regarding this, an appropriate analysis that we

use in this work is provided in Ref. [88] for the ellipsoidal-

collapse dark matter halos.

Moreover, the main challenge regarding the statistics

governing dark matter halos is their mass distribution in

the cosmic web. Fortunately, in providing a proper sta-

tistical description of dark matter halos, a function called

the halo mass function can be introduced [90]. The halo

mass function is a powerful probe in cosmology to classify

the mass of dark matter halo structures. In other words,

the halo mass function determines the number density of

dark matter halos depending on their local quantities. In

Ref. [91], a suitable description of the scaled diﬀerential

mass function is presented as

dM =g(σ)ρm

dln(σ−1)

dM ,(4)

where ρmrepresents the cosmological matter density,

n(M) speciﬁes the number density of dark matter halos

with mass M,σ(M, z) is the variance of linear overden-

sities on mass Mand redshift z, and g(σ) is the ﬁtting

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OntheMergerRateofPrimordialBlackHolesinCosmicVoidsSaeedFakhry,1,2,SeyedSajadTabasi,3,2,yandJavadT.Firouzjaee4,5,2,z1DepartmentofPhysics,ShahidBeheshtiUniversity,Evin,Tehran19839,Iran2PDATLaboratory,DepartmentofPhysics,K.N.ToosiUniversityofTechnology,P.O.Box15875-4416,Tehran,Iran3DepartmentofPhysics...

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On the Merger Rate of Primordial Black Holes in Cosmic Voids Saeed Fakhry1 2Seyed Sajad Tabasi3 2yand Javad T. Firouzjaee4 5 2z 1Department of Physics Shahid Beheshti University Evin Tehran 19839 Iran

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