FRW model with two-fluid source in fractal cosmology

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Pramana – J. Phys. (2022) 96:133 © Indian Academy of Sciences

https://doi.org/10.1007/s12043-022-02364-5

FRW model with two-ﬂuid source in fractal cosmology

DDPAWAR

1,∗,DKRAUT

2and W D PATIL3

1School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded 431 606, India

2Department of Mathematics, Shivaji Mahavidyalaya, Renapur, Latur 413 527, India

3Department of Applied Mathematics, A.C. Patil College of Engineering, Navi Mumbai 410 210, India

∗Corresponding author. E-mail: dypawar@yahoo.com

MS received 1 September 2021; revised 18 December 2021; accepted 7 February 2022

Abstract. The present paper deals with ﬂat Friedmann–Robertson–Walker (FRW) model with two-ﬂuid sources

in fractal cosmology. One ﬂuid in this model represents the Universe’s matter content, while the other is a radiation

ﬁeld that models the cosmic microwave background. To obtain the deterministic model, we used the pressure–

density relationship for matter via the gamma law equation of state pm=(γ −1)ρ

m,1≤γ≤2. The solutions

of fractal ﬁeld equations are obtained in terms of Kummer’s conﬂuent hypergeometric function of the ﬁrst kind.

Some physical parameters of the models are obtained and their behaviour is discussed in detail using graphs.

Keywords. Fractal cosmology; two-ﬂuid; Friedmann–Robertson–Walker model.

PACS Nos 98.80.-k; 04.50.Kd; 98.80.Cq

1. Introduction

The Friedmann–Robertson–Walker (FRW) model,

which is homogeneous and isotropic, is thought to be

a good approximation of the early and current stages

of the Universe. The 2.73K CMBR has attracted many

researchers to study the FRW model with two ﬂuids [1–

3]. In literature it is found that rather than using two

sequences of a single ﬂuid model, a cosmic evolution

based on two-ﬂuid models is preferable. In two-ﬂuid

models, one ﬂuid dominates the other, and both ﬂu-

ids are assumed to be present throughout the cosmic

evolution. Such models are used to describe the evolu-

tion of the Universe from a radiation-dominated phase

to a matter-dominated phase. The radiation and matter

ﬂuid models are cosmologically important. Their appli-

cations to cosmology are inspired by observations [4]

that recommend that the radiation frame and therefore

the matter frame of the Universe might not coincide.

In this work, we consider a homogeneous and isotropic

FRW Universe with two ﬂuid sources within the

framework of fractal cosmology. Many researchers

are interested in studying the two-ﬂuid cosmological

models in general relativity as well as in alternative

theories of gravity. Goswami and Pradhan [5]have

recently studied a two-ﬂuid cosmological model in a

Bianchi type-I Universe. Rao et al [6] investigated

homogeneous Kantowski–Sachs two-ﬂuid rotating cos-

mological model in Brans–Dicke theory of gravitation.

Chimento [7] investigated interacting ﬂuids that produce

identical, dual and phantom cosmologies. In general

relativity, Samanta [8] examined two-ﬂuid anisotropic

Bianchi type-III cosmological model with variable G

and . Pawar and Solanke [9] studied the two-ﬂuid dark

energy cosmological model in the Saez Ballester the-

ory of gravity. Adhav et al [10] studied the two-ﬂuid

cosmological models in Bianchi type-III space–time. In

the theory of general relativity and the Lyra manifold,

a bulk viscous ﬂuid with magnetised string and plane

symmetric string cosmological models has been stud-

ied by Pawar et al [11–13]. Mishra et al [14] studied the

cosmological model of an accelerating dark energy in

two ﬂuids, with a mixed scale factor. Katore and Hatkar

[15] have investigated FRW Universe in the framework

of f(R)gravity with two co-moving perfect ﬂuids as

the source of gravitational ﬂuid. Singh et al [16]have

explored rotating anisotropic two-ﬂuid Universe cou-

pled with radiation and a scalar ﬁeld. Tiwari et al [17,18]

have analysed EoS parameter for dark energy in FRW

and Bianchi type-III space–time ﬁlled with dark energy

and barotropic ﬂuid. Pawar and Dagwal [19] studied two

ﬂuid sources in F(T)theory of gravitation. Mishra et al

[20] studied stability of the two ﬂuid cosmological mod-

els. Hasmani and Pandya [21] investigated cosmological

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133 Page 2 of 10 Pramana – J. Phys. (2022) 96:133

models for the static spherically symmetric space–time

with charged anisotropic ﬂuid distribution in the context

of Rosen’s bimetric general relativity (BGR).

Recently, fractals have been used to study the Uni-

verse. Mandelbrot [22] developed the concept of fractals

and proposed that the standard cosmological principle

be replaced by a conditional cosmological principle.

Pietronero [23] explained the fractal distribution of

galaxies to precisely ﬁt astronomical data. On the scale

of relativity, Laurent [24] proposed the fractal nature

of space–time. Eternally existing chaotic inﬂationary

model of the Universe was investigated by Linde [25].

Calcagni [26] investigated the ﬂat FRW model within

a fractal framework. In fractal theory of gravitation,

Mustafa Salti et al [27] investigated the extended ver-

sions of holographic and Ricci dark energy models.

Karami et al [28] studied the ﬂat fractal FRW Universe

in dark energy and dark matter. In fractal cosmology,

Hosseienkhani et al [29] investigated the anisotropic

behaviour of dark energy. Aly and Selim [30] studied

f(T)-modiﬁed gravity model for the dark energy in

fractal Universe. Recently, Pawar et al [31] studied FRW

space–time within the domain wall in the framework of

fractal cosmology.

Motivated by the aforementioned studies, the current

paper attempts to investigate the ﬂat FRW model with

two ﬂuids within the framework of fractal cosmology.

We assumed the fractal parameter ω= 0andthepower

law form of the fractal function vto obtain the solutions

of fractal ﬁeld equations. The analysis was carried out

in the UV regime with no cosmological constant. In §2,

we have presented the formalism of fractal gravity. In

§3, we obtained the fractal ﬁeld equations. Solutions of

ﬁeld equations under the inﬂuence of fractal cosmology

are derived in §4. Section 5deals with a brief discussion

of the results. Finally, §6highlights our conclusions.

2. Formalism of fractal gravity

Calcagni [26] proposed a fractal cosmological model

for power counting renormalisable ﬁeld theory living in

fractal space–time. The action in this model is Lorentz

invariant and equipped with Lebesue–Stieltjes mea-

sure. The total action of Einstein’s gravity in fractal

space–time, assuming matter is minimally coupled with

gravity, can be written as [26]

S=Sg+Sm,(1)

where Sgis the gravitational part of the action

Sg=M2

2√−g(R−2−ω∂av∂

av) dξ(x)(2)

and Smis the matter part of the action

Sm=√−gL

mdξ(x). (3)

Here M2=1/8πG,,R,Lmand gdescribe the

reduced Planck mass, cosmological constant, Ricci cur-

vature scalar, Lagrangian density of the matter part

of action and determinant of dimensionless metric gab

respectively. The quantities ωand vare fractal parameter

and fractal function respectively. The standard measure

d4xin the action is replaced by Lebesgue–Stieltjes mea-

sure dξ(x)and dξ(x)=(d4x)v(x).

The Einstein’s equations in fractal Universe are

derived by varying total action (1) with respect to the

metric tensor gab

Rab −1

2gab(R−2) +gab

v

v−abv

+1

2gab∂σv∂

σv−∂av∂

bv=8πGT

ab,(4)

where Tab is the energy–momentum tensor given by

Tab =−2

√−g

δsm

δgab (5)

and =

μμis the covariant derivative.

3. Metric and the ﬁeld equations in fractal

cosmology

Consider the FRW space–time in the form

ds2=−dt2+S2(t)

×dr2

1−kr2+r2(dθ2+sin2θdφ2),(6)

where the scale factor S(t)is a function of cosmic time

t,kis a constant which denotes the spatial curvature

and k=1,0,−1 represents closed, ﬂat and open FRW

Universe, respectively.

The spatial volume (V), Hubble parameter (H)and

expansion scalar (θ) for metric (6) are respectively given

V=S3,H=˙

S,θ=3˙

S,(7)

where the overhead dot denotes the derivative with

respect to cosmic time (t).

The deceleration parameter is deﬁned as

q=−S¨

S2.(8)

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Pramana–J.Phys.(2022)96:133©IndianAcademyofScienceshttps://doi.org/10.1007/s12043-022-02364-5FRWmodelwithtwo-ﬂuidsourceinfractalcosmologyDDPAWAR1,∗,DKRAUT2andWDPATIL31SchoolofMathematicalSciences,SwamiRamanandTeerthMarathwadaUniversity,Nanded431606,India2DepartmentofMathematics,ShivajiMahavidyalaya,...

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FRW model with two-fluid source in fractal cosmology

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