Constraining Anisotropic Cosmological Model in fRLmGravity N. S. Kavya 1V . Venkatesha 1 Sanjay Mandal 2 and P .K. Sahoo2 1Department of P.G. Studies and Research in Mathematics

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Constraining Anisotropic Cosmological Model in f(R,Lm)Gravity

N. S. Kavya ,1, ∗V. Venkatesha ,1, †Sanjay Mandal ,2, ‡and P.K. Sahoo 2, §

1Department of P.G. Studies and Research in Mathematics,

Kuvempu University, Shankaraghatta, Shivamogga 577451, Karnataka, INDIA

2Department of Mathematics, Birla Institute of Technology and Science-Pilani,

Hyderabad Campus, Hyderabad 500078, INDIA

(Dated: October 19, 2022)

The observational evidence regarding the present cosmological aspects tells us about the presence

of very little anisotropy in the universe on a large scale. Here, in this paper, we attempt to study

locally rotationally symmetric (LRS) homogeneous Bianchi-I spacetime with the isotropic matter dis-

tribution. This is done within the framework of f(R,Lm)gravity. Particularly, we consider a non-

linear f(R,Lm)model, f(R,Lm) = 1

2R+Lα

m. Furthermore, ω, the equation of state parameter,

which is vital stuff in determining the present phase of the universe is constrained. To constrain the

model parameters and the equation of state parameter, we use 57 Hubble data points and 1048 Pan-

theon supernovae type Ia data sample. And, for our statistical analysis, we use Markoc Chain Monte

Carlo (MCMC) simulation. Moreover, with the help of obtained values of parameters, we measure

the anisotropy parameter for our model.

Keywords: Equation of state parameter, f(R,Lm)gravity, observational constraints, anisotropy

parameter.

I. INTRODUCTION

Over the past few decades, many scientiﬁc explo-

rations have been taking place to decipher the mys-

tic behavior of the universe. Right from the early

time inﬂation to the late time acceleration, from the

black holes to the wormholes, from the dark energy to

the gravitational waves, their entire course has been

probing the very nature of the universe. Just to look

into the cosmological principle, the universe on a large

scale, was presumed to be both isotropic and homo-

geneous. But in 1992, Cosmic Background Explorer

(COBE) successfully made a signiﬁcant assertion about

the existence of a small anisotropy in the large-scale

cosmic microwave background [1]. Moreover, in the

later years, this was further supported by the measure-

ments made by Balloon Observations of Millimetric Ex-

tragalactic Radiation and Geophysics (BOOMERanG)

[2], Cosmic Background Imager (CBI) [3], Wilkinson Mi-

crowave Anisotropy Probe (WMAP)[4], and the Plank

collaborations[5]. Furthermore, intriguing advance-

ments in the ﬁeld of cosmology took place through the

observational results of the two teams led by Perlmutter

and Riess [6,7]. These studies strive to endorse that the

universe is currently in the phase of accelerated expan-

∗kavya.samak.10@gmail.com

†vensmath@gmail.com

‡sanjaymandal960@gmail.com

§pksahoo@hyderabad.bits-pilani.ac.in

sion. To this point, there arose a question regarding the

isotropic nature of the expansion of the universe. Inter-

estingly, recent developments suggest that the universe

tends to expand at a different rate in different directions

[8]. Though FLRW cosmology is most successful, it is

built based on cosmological principles. However, the

observational evidence attempts to elucidate the pres-

ence of a slight difference in the strengths of microwaves

coming from different axes. For this reason, the space-

time that can appropriately describe anisotropic and

homogeneous geometry is Bianchi cosmology. Several

works on such Bianchi cosmology with different modi-

ﬁed gravity frameworks can be found in the literature.

(See ref [9–20])

In the present scenario, to deal with the study of such

aspects, the modiﬁed theoretic approach sounds more

potent. Among these, the f(R)theory of gravity has

produced a reliable framework for evaluating the cur-

rent cosmic evolution [21]. Indeed, f(R)theories can

adequately explain the interpretations of late-time accel-

eration [20,22], the exclusion of the dark matter entity

in the analysis of the dynamics of massive test parti-

cles [23], and the uniﬁcation of inﬂation with dark en-

ergy [24]. Furthermore, numerous justiﬁcations indicate

that the higher-order theories, like f(R)gravity, are ca-

pable of explaining the ﬂatness of galaxies’ rotational

curves [25]. With these motivations, several coupling

theories came into existence [26–28]. One such theory

is the f(R,Lm)theory of gravity [29]. Notably, this fa-

vors the occurrence of an extra force that is orthogonal to

arXiv:2210.09307v1 [gr-qc] 17 Oct 2022

four-velocity. In addition, the so-called ‘extra’ force ac-

counts for the non-geodesic motions of the test particle.

Consequently, a violation of the equivalence principle

can be observed. Numerous contributions to this theory

can be seen in the literature [30–38]. Recently, Jaybhaye

et al have studied cosmology in f(R,Lm)gravity [39].

In the present work, we center on the study of the-

oretical exploration and observational validation of the

LRS Bianchi type I spacetime and effectuate this in terms

of f(R,Lm)formalism. Moreover, in assessing the ex-

panding universe the equation of state parameter plays

a prominent role. This predicts the ﬂuid type in space-

time. In our work, we emphasize constraining this cos-

mological parameter ωand obtaining the best ﬁt values

as per the observational measurements. This is accom-

plished with a statistical approach for incorporated sets

of data samples. We use two types of data samples such

as Hubble measurements and Pantheon SNe Ia sample

. Further, with the anisotropy parameter, we measure

anisotropy in spacetime.

This manuscript is organized as follows: in section II,

the basic formulation of f(R,Lm)gravity is presented.

The analysis of LRS Bianchi I within the framework of

the f(R,Lm)gravity is made in section III. Section IV

is brought with the examination of observational con-

straints and discussion of results. Finally, the last section

V, gives some concluding remarks.

II. THE BASIC FIELD EQUATIONS IN f(R,Lm)

GRAVITY

With the matter lagrangian density Lmand the Ricci

scalar R, the action integral for f(R,Lm)theory reads,

S=Zf(R,Lm)p−g d4x, (1)

where frepresents an arbitrary function of Rand Lm.

The ﬁeld equation for the f(R,Lm)gravity [29], ob-

tained by varying the action integral (1) with respect to

the metric tensor gµν is given by,

fR(R,Lm)Rµν + (gµν∇µ∇µ− ∇µ∇ν)fR(R,Lm)

−1

2f(R,Lm)−fLm(R,Lm)Lmgµν =

2fLm(R,Lm)Tµν.

(2)

Here, fR(R,Lm)≡∂f(R,Lm)

∂R,fLm(R,Lm)≡∂f(R,Lm)

∂Lm,

and Tµν is the Energy-Momentum Tensor (EMT) that

can be expressed as,

Tµν =−2

√−g

δ(√−gLm)

δgµν =gµνLm−2∂Lm

∂gµν . (3)

Now, from the explicit form of the ﬁeld equation (2),

the covariant divergence of EMT Tµν can be obtained as,

∇µTµν =2n∇µln fLm(R,Lm)o∂Lm

∂gµν . (4)

Furthermore, on contracting the ﬁeld equation (2) we

get,

3∇µ∇µfR(R,Lm) + fR(R,Lm)R − 2f(R,Lm)

−fLm(R,Lm)Lm=1

2fLm(R,Lm)T.(5)

By considering the above equation, the relation be-

tween the trace of EMT T=Tµ

µ,Lmand Rcan be es-

tablished.

III. LRS BIANCHI-I COSMOLOGY IN f(R,Lm)

GRAVITY

For anisotropic and spacially homogeneous LRS

Bianchi-I spacetime, the metric is described by,

ds2=−dt2+A2(t)dx2+B2(t)dy2+B2(t)dz2, (6)

where, Aand Bare metric potentials that are the func-

tions of (cosmic) time talone. If A(t) = B(t) = a(t),

then one can analyze the scenarios in ﬂat FLRW space-

time. Now, the Ricci scalar for LRS Bianchi-I spacetime

can be expressed as,

R=2"¨

A+2¨

B+2˙

A˙

AB +˙

B2#(7)

With the directional Hubble parameters Hx,Hyand

Hz, the Ricci scalar for the corresponding metric is given

by,

R=2(˙

Hx+2˙

Hy) + 2(H2

x+3H2

y) + 4HxHy. (8)

Here, Hx=˙

Aand Hy=˙

B=Hzindicate the direc-

tional Hubble parameters along the corresponding co-

ordinate axes. For Hx=Hy=H, i.e., for FLRW cosmol-

ogy, the equation R=2(˙

Hx+2˙

Hy) + 2(H2

x+3H2

y) +

4HxHyreduces to R=6(2H2+˙

H). In the present work,

we are supposing the matter distribution to be described

by the energy-momentum tensor of a perfect ﬂuid,

Tµν = (ρ+p)UµUν+p gµν, (9)

where ρis the energy density and pis the pressure. The

four-velocity, Uµsatisﬁes the condition UµUµ=−1 and

UµUµ;ν=0. Thus, the ﬁeld equation (2) takes the form,

−˙

fR(Hx+2Hy) + ( ˙

Hx(t) + 2˙

Hy+H2

x+2H2

y)fR

−1

2f−fLmLm=−ρfLm

(10)

−¨

fR−2Hy˙

fR+ ( ˙

Hx+H2

x+2HxHy)fR

−1

2f−fLmLm=pfLm

2,(11)

−¨

fR−2Hx˙

fR+ ( ˙

Hy+2HxHy)fR

−1

2f−fLmLm=pfLm

2.(12)

The dot (·)here represents the derivative with respect

to the time tand f≡f(R,Lm).

Further, one can express the spatial volume Vof the

spacetime as,

V=a3=AB2. (13)

Thus the mean value of the Hubble parameter is given

by,

H=˙

a=1

3(Hx+2Hy). (14)

In further study, we are going to investigate the physical

cosmological model and their application in the context

of f(R,Lm)gravity using the above set of equations.

A. Physical Model:

In the present study, we shall focus on the cosmo-

logical aspects of f(R,Lm)theory, with the relation be-

tween Rand Lmbeing

f(R,Lm) = 1

2R+Lα

m, (15)

where, α6=0 is a model parameter and one can retain

GR for α=1.

Now, to ﬁnd an exact solution to the ﬁeld equations

(10)-(12), we have to consider the constraining relation.

To this point, we shall presume the anisotropic relation

that can be written in terms of shear (σ)and expansion

scalar (θ)as,

θ2∝σ2,

so that, for constant σ

θ, the Hubble expansion can

achieve isotropy [42,43]. This condition gives rise to

A(t) = B(t)n, (16)

for some real non-zero n, and for n=1, we can retrieve

ﬂat FLRW cosmology. With this, one can get the relation

between directional Hubble parameters as,

Hx=nHy. (17)

Therefore, averaged Hubble parameter can takes the

form,

H=n+2

3Hy. (18)

Additionally, we shall relate the the pressure pand en-

ergy density ρby,

p=ωρ. (19)

Now, we have two choices for the Lagrangian to pro-

ceed further such as Lm=−ρor Lm=p[40]. But, in

our study we consider Lm=−ρbecause it is the most

adequate choice presented in [40]. In literature, these

are many studies which have been explored the choices

for Lmand their applications [to see more details please

check the references therein [40,41].

Applying the above conditions, the ﬁeld equations

(10)-(12) become,

9(2n+1)

(n+2)2H2=−(−ρ)α, (20)

6˙

(n+2)+27H2

(n+2)2=−(−ρ)αα(1+ω) + 1,

(21)

3˙

Hn+1

n+2+9H2n2+3

(n+2)2=−(−ρ)αα(1+ω) + 1.

(22)

With the help of aforementioned equations we can ob-

tain an expression for the Hubble parameter Hin terms

of redshift zas,

H(z) = γ1(z+1)γ1/γ2, (23)

where, γ1=2(n+2)and γ2=−3[α(ω+1)(2n+1) +

2(n−1)]. Here, we used the scale factor a(t)and red-

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ConstrainingAnisotropicCosmologicalModelinf(R,Lm)GravityN.S.Kavya,1,V.Venkatesha,1,SanjayMandal,2,andP.K.Sahoo2,§1DepartmentofP.G.StudiesandResearchinMathematics,KuvempuUniversity,Shankaraghatta,Shivamogga577451,Karnataka,INDIA2DepartmentofMathematics,BirlaInstituteofTechnologyandScience-Pilani,H...

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Constraining Anisotropic Cosmological Model in fRLmGravity N. S. Kavya 1V . Venkatesha 1 Sanjay Mandal 2 and P .K. Sahoo2 1Department of P.G. Studies and Research in Mathematics

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