Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion Arnab Sarkar1Amna Ali2yK. Rajesh Nayak3zand A. S. Majumdar1x

2025-05-06 0 0 437.05KB 9 页 10玖币

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Enhanced power of gravitational waves and rapid coalescence of black hole binaries

through dark energy accretion

Arnab Sarkar,1, ∗Amna Ali,2, †K. Rajesh Nayak,3, ‡and A. S. Majumdar1, §

1Department of Astrophysics and High Energy Physics, S. N. Bose National Centre for Basic Sciences,

JD Block, Sector III, Salt lake city, Kolkata-700106, India

2RsRL, Dubai, United Arab Emirates

3Indian Institute of Science Education and Research (IISER), Mohanpur, West Bengal-741246, India

(Dated: October 25, 2022)

Abstract

We consider the accretion of dark energy by constituent black holes in binary formations during the

present epoch of the Universe. In the context of an observationally consistent dark energy model,

we evaluate the growth of black holes’ masses due to accretion. We show that accretion leads to

faster circularization of the binary orbits. We compute the average power of the gravitational waves

emitted from binaries, which exhibits a considerable enhancement due to the eﬀect of growth of

masses as a result of accretion. This in turn, leads to a signiﬁcant reduction of the coalescence

time of the binaries. We present examples pertaining to various choices of the initial masses of the

black holes in the stellar mass range and above, in order to clearly establish a possible observational

signature of dark energy in the emerging era of gravitational wave astronomy.

I. INTRODUCTION

After the ﬁrst direct detection of gravitational waves from

a merging binary of black holes by aLIGO [1], and subse-

quent series of detections from similar sources [2–4], a new

era in observational astronomy has begun. Besides binaries

of compact objects in bounded orbits, there are various other

mechanisms of production of gravitational waves from a wide

varieties of sources, such as nearby ﬂy-pass of two compact

objects in unbounded orbits [5], gravitational collapse of suﬃ-

ciently massive stars [6], cosmological phase transitions [7,8],

breaking of cosmic strings [9,10], inﬂation and pre-heating

[11,12] etc.. However, till date the observations by the aLIGO

and VIRGO detectors have been carried out from mainly one

type of sources, which are the binaries of compact objects,

viz., black holes and neutron stars.

Gravitational wave observations have been used to esti-

mate and constrain various astrophysical and cosmological

parameters associated with the generation and propagation

of these gravitational waves. Among these, some important

ones worth mentioning are: (i) estimating the Hubble param-

eter [13,14], (ii) constraining a large class of cosmological

scalar-tensor theories [15,16], (iii) constraining the mass of

gravitons for bimetric-gravity theories [17], (iv) investigating

the state of matter inside a neutron star [18], (v) constraining

higher-dimensional theories [19], and there are several others.

Attempts to constrain dark energy, responsible for the accel-

erated expansion of the late Universe [22], have been made in-

directly using gravitational wave observations, either through

the estimation of the Hubble parameter, or through constrain-

ing cosmological scalar-tensor theories.

Late time acceleration of expansion of the Universe is one

of the most intriguing discoveries of recent times, which was

∗arnabsarkar@bose.res.in, arnab.sarkar14@gmail.com

†amnaalig@gmail.com

‡rajesh@iiserkol.ac.in

§archan@bose.res.in

directly conﬁrmed from supernovae Ia observations in 1998

[20,21] and was also supported by various indirect probes.

Many theoretical approaches have been employed to explain

the current cosmic acceleration. The component of the Uni-

verse providing the required negative pressure for driving this

accelerated expansion is generically called ‘dark energy’ [23].

As normal matter (radiation, baryonic matter or cold dark

matter) is gravitationally attractive, the standard lore is to

assume the presence of a relativistic ﬂuid which is repulsive

in nature, as the dark energy candidate. The simplest candi-

date of dark energy is the cosmological constant Λ, which is

mostly consistent with cosmological observations. However, it

is plagued with conceptual problems, for example, ﬁne-tuning

and coincidence problems [24], which are theoretical in na-

ture. With a hope to address these problems, cosmologists

have proposed mechanisms where the role of dark energy is

played by a completely diﬀerent component of the Universe,

which may have a variable equation-of-state parameter.

Many varieties of dark energy models have been pro-

posed, theoretically studied and observationally constrained

till now. There exist a wide class of scalar ﬁeld models cou-

pled to gravity. Among these, minimally coupled ones, called

quintessence, in which cosmic acceleration is driven by the

potential energy [25,26], are known to alleviate some of the

problems of the cosmological constant. Scalar ﬁelds, in which

the cosmic acceleration is driven by the kinetic energy, called

‘k-essence’ [27–29], have also been studied, motivated from

uniﬁcation and quantum gravity scenarios. Such models may

further yield a consistent picture of the complete evolution,

starting from the early era inﬂation, the subsequent dark mat-

ter domination, and ﬁnally the late time acceleration [30,31].

Other alternatives include random barotropic ﬂuids with pre-

determined forms of the equation-of-state parameter, such as

the Chaplygin gas models [32], string theory motivated mod-

els [33–35] and braneworld models [36,37]. There also exist

approaches without requiring additional ﬁelds[38–41].

A major diﬀerence between the scalar ﬁeld and other ﬂuid

models of dark energy with the ΛCDM model (and other ap-

proaches not requiring additional ﬁelds) is that the former

type of dark energy is subjected to accretion by the black

arXiv:2210.12502v1 [gr-qc] 22 Oct 2022

holes present in the Universe. In fact, those back holes with

surroundings containing insuﬃciently available other forms of

matter-energy for accretion, would still accrete the scalar ﬁeld

dark energy, which is uniformly distributed throughout the

Universe. Accretion of various types of dark energy by black

holes has been a subject of theoretical interest for a consid-

erable time [42–46]. On the basis of various works done till

date, it is widely accepted that the mass of a black hole would

increase due to steady spherical accretion if the equation-of-

state parameter of the dark energy wis >−1. On the other

hand, accretion would result in mass loss of a black hole, for

phantom type dark energy with w < −1.

If dark energy exists in the Universe in a form which can

be accreted by black holes, the result would not be limited to

just the change of masses of the black holes. It is expected

that other phenomena associated with the black holes would

also be inﬂuenced. The evolution of binaries formed with

the black holes, the gravitational wave emitted from those

binaries and coalescence of those binaries are some of the

physical processes which get directly aﬀected if the masses of

the concerned black holes are changing continuously instead of

being constant. The eﬃcacy of the above eﬀects, in particular,

whether the modiﬁed variation of gravitational energy of the

binary system could be detectable via the rate of change of

the orbital radius, has been a subject of debate [47,48] in the

case of spherically symmetric accretion of dark energy [42].

In the present work our motivation is to explore the prob-

lem of associated modiﬁcation of black hole binary param-

eters due to accretion in the context of a popular k-essence

model of dark energy. Speciﬁcally, we consider a string theory

[51] inspired low energy eﬀective action framework contain-

ing a dilaton scalar ﬁeld [52]. The resultant k-essence dark

energy scenario [53] is compatible with cosmological observa-

tions [54]. Here we study spherical accretion of the k-essence

dilatonic ghost condensate dark energy by black holes. This

falls within a class of models known as ‘ghost condensates’

[55]. Considering binary formations of black holes in the early

inspiral stage, we study main aspects of evolution of the or-

bits, due to continuous change of masses of the component

black holes of such binaries, resulting from spherical accretion

of the chosen model of k-essence dark energy. More speciﬁ-

cally, we study the pace of shrinking of such an orbit and

the average power of the gravitational wave emitted from the

orbit in the course of its evolution, and perform quantitative

comparisons of the diﬀerences with the case when the masses

of component black holes are constant. We further investigate

the modiﬁcation in the time required to reach the coalescence

stage of such a binary, in comparison with the constant mass

case.

The paper is organised as follows. In section II, we study

the growth of black hole mass due to accretion of the chosen

model of k-essence dark energy in the late Universe. In sec-

tion III, we investigate the eﬀect of growing masses of black

holes on the evolution of binaries. We compute the rate of de-

crease of orbital radius after circularization of orbits, and the

average power of the emitted gravitational waves. We com-

pare these results with the case of binaries with black holes of

constant masses without accretion. In section IV, we estimate

the reduction in the time required for reaching coalescence-

stage by such binaries. We present our concluding remarks in

section V.

II. DARK ENERGY ACCRETION BY BLACK

HOLES IN A K-ESSENCE MODEL

K-essence scalar ﬁelds are the dynamical dark energy mod-

els where the acceleration is driven by kinetic term in the

scalar ﬁeld Lagrangian. Among many k-essence models, we

choose a particular string-inspired ghost condensate model,

called ‘k-essence dilatonic ghost condensate’, which can suc-

cessfully describe the cosmological evolution, while simulta-

neously satisﬁes the necessary conditions of quantum stability

and sound speed [53,56]. This model has also found to be

observationally consistent [54].

The condition on sound speed for any scalar ﬁeld dark energy

model is simply that the sound speed can not exceed the speed

of light in vacuum (c) i.e. it can not have super-luminal speed.

In this regard, it is worthwhile to mention that the sound

speed makes an important diﬀerence between quintessence

and k-essence models. While for standard quintessence mod-

els, with canonical scalar ﬁelds, the sound speed is always

equal to the speed of light ; for the k-essence models it is

not so. This fact of having varying sound speeds through the

cosmic evolution gives various ways of distinguishing diﬀerent

k-essence models from one-another and from the quintessence

models [57]. In fact this diﬀerence of sound speed of k-essence

models with the quintessence models is one of the main rea-

sons, for which we have chosen a k-essence model for this

study.

The action of k-essence scalar ﬁeld ϕ, along with non-

relativistic matter and radiation, can be generally written as

[27] :

S=Zd4x√−g1

2κ2R+L(ϕ, X)+Sm,(1)

where κ= (8πG/3)1/2,Ris the Ricci-scalar and Lis a func-

tion of the k-essence scalar ﬁeld ϕand its kinetic energy

X=−(1/2)gµν ∂µϕ∂νϕ.Smis the action contributed from

the non-relativistic matter and radiation. In case of the spe-

ciﬁc model considered here, the Lagrangian density is given

by [53,56]:

L=−X+eκλϕ X2

M4,(2)

where Mis a constant having the dimension of mass and λ

is a constant dimensionless parameter, which is set according

to stability conditions.

The set of equations governing the cosmological dynamics

of this k-essence model can be conveniently written in terms

of three dimensionless parameters: [53,56] :

x1=κ˙ϕ

√6H, x2=ϕ2eκλϕ

2M4, x3=κ√ρr

√3H,(3)

where His the Hubble-parameter and ρris the density of

radiation in the Universe. With these dimentionless parame-

ters x1, x2and x3, the evolution equations can be cast in the

following autonomous form:

dx1

dN =−x1

6(2x2−1) + 3√6λx1x2

(6x2−1)

+x1

2(3 −3x2

1+ 3x2

1x2+x3),

(4)

dx2

dN =x2

3x2(4 −√6λx1)−√6(√6−λx1)

1−6x2

,(5)

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EnhancedpowerofgravitationalwavesandrapidcoalescenceofblackholebinariesthroughdarkenergyaccretionArnabSarkar,1,AmnaAli,2,yK.RajeshNayak,3,zandA.S.Majumdar1,x1DepartmentofAstrophysicsandHighEnergyPhysics,S.N.BoseNationalCentreforBasicSciences,JDBlock,SectorIII,Saltlakecity,Kolkata-700106,India2RsRL,...

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Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion Arnab Sarkar1Amna Ali2yK. Rajesh Nayak3zand A. S. Majumdar1x

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