An fRT Gravity Based FLRW Model and Observational Constraints Anirudh Pradhan1 Gopikant Goswami2 Rita Rani3 Aroonkumar Beesham45 1Centre for Cosmology Astrophysics and Space Science CCASS GLA University Mathura-281 406 Uttar

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An f(R,T) Gravity Based FLRW Model and Observational Constraints

Anirudh Pradhan1, Gopikant Goswami2, Rita Rani3, Aroonkumar Beesham4,5

1Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura-281 406, Uttar

Pradesh, India

2,3Department of Mathematics, Netaji Subhas University of Technology, Delhi, India

4Department of Mathematical Sciences, University of Zululand Private Bag X1001 Kwa-Dlangezwa 3886 South

Africa

5Faculty of Natural Sciences, Mangosuthu University of Technology, P O Box 12363, Jacobs 4052, South Africa

1E-mail: pradhan.anirudh@gmail.com

2E-mail: gk.goswami9@gmail.com

3E-mail: rita.ma19@nsut.ac.in

4,5E-mail: abeesham@yahoo.com

Abstract

We attempt to construct a Friedmann-Lemaitre-Robertson-Walker(FLRW) cosmological model in f(R, T )

gravity which exhibits a phase transition from deceleration to acceleration at present. We take f(R, T ) =

R+ 2λT ,λbeing an arbitrary constant. In our model, the λparameter develops a negative pressure in the

universe whose Equation of state is parameterized. The present values of model parameters such as density,

Hubble, deceleration, Equation of state, and λare estimated statistically by using the Chi-Square test. For this,

we have used three diﬀerent types of observational data sets: the 46 Hubble parameter data set, the SNeIa 715

data sets of distance modulus, and the 66 Pantheon data set (the latest compilation of SNeIa 40 bined plus

26 high red shift apparent magnitude mbdata set in the red shift ranges from 0.014 ≤z≤2.26). We have

calculated the transitional red shift and time. The estimated results for the present values of various model

parameters are found as per expectations and surveys. Interestingly, we get the present value of the density ρ0,

≃1.5ρc. The critical density is estimated as ρc≃1.88 h2

010−29 gm/cm3in the literature. The higher value of

the present density is attributed to the presence of some additional energies in the universe apart from baryon

energy. We have examined the behavior of the pressure in our model. It is negative and produces acceleration

in the universe. Its present value is obtained as p0≃ −0.7ρ0.

Keywords:f(R, T ) theory; FLRW metric; Observational parameters; Transit universe;

Observational constraints

PACS number: 98.80-k, 98.80.Jk, 04.50.Kd

1 Introduction

ΛCDM cosmological model is the traditional concordance model which ﬁts best with the latest observational

constraints despite its failure to explain ﬁne-tuning and the cosmic coincidence problems ([1] −[4]). A set of

observational results are given in references ([5] −[29]) which expresses the fact that our universe is accelerating.

To explain this, a large amount of anti-gravitational and repulsive energy given the name “exotic dark energy

(DE)” is believed to be present in the universe and this DE is responsible for the acceleration. In observational

cosmology surveys, there are searches for mainly four parameters: Hubble parameter (H0), distance modulus (µ),

apparent magnitude (mb), and the deceleration parameters (DP - q0). So, these parameters are important tools

to model a physical universe. The traditional Friedmann-Lemaitre-Robertson-Walker (FLRW) model is so far the

best-ﬁt model which describes a homogeneous and isotropic universe. It originates with a big bang singularity,

arXiv:2210.15433v3 [gr-qc] 2 Jun 2023

then sudden inﬂation cools down its heavy contents to permit the production of sub-atomic and quantum particles.

Thereafter the universe enters into the radiation and matter-dominated eras. But it fails to explain the higher

value of the density of the universe. It also does not explain why the SNIa supernovae are more distant than

expected, which requires an acceleration in the universe instead of deceleration as predicted by the FLRW model.

There are two schools of thought to explain and analyze these anomalies. In the former one ([30] −[35]), it

is assumed that along with baryon matter, DE exists producing negative pressure. As a result, it repels matter,

thus producing acceleration in the universe. DE is discussed in the framework of general relativity. The second

school of thought is based on the theme that nonlinear curvature may develop a geometry that could change the

dynamics of matter to produce an acceleration in the universe. This requires modiﬁcations in Einstein’s ﬁeld

equations. A group headed by A A Stravinsky, Antonio De Felice, and Tsujikawa et al. ([36]−[64]) modiﬁed

Einstein ﬁeld equations by replacing the Ricci scalar Rwith an arbitrary function of the Ricci scalar Rand the

energy-momentum tensor Tij in the Einstein Hilbert action, and formulated modiﬁed theories of gravitation. Their

views are simple in the sense that matter creates gravitation and gravitation creates curvature. Curvature will

not remain silent, it should also act on matter to produce some dynamic results. Accordingly, so many modiﬁed

theories of gravity f(R), f(R, G), f(R, T ) gravity, f(R, T ϕ) and many more have surfaced in the literature. Out

of this f(R, T ),is one of the popular options.

In the present work, we attempt to model a universe with reference to the present context in the framework of an

FLRW space-time metric using the ﬁeld equations of f(R, T ) gravity. The propagator of the theory has suggested

three options for the speciﬁc functional form of f(R, T ). We consider the ﬁrst popular one f(R, T ) = R+ 2f(T),

where we have taken f(T) = λT and λis an arbitrary parameter. The aim is to develop an accelerating model.

For this, it is proposed that the λparameter is associated with negative pressure, and the equation of state(EoS)

(ω) is parameterized as per Gong and Zhang( [61]). Like the Einstein ﬁeld equations for an FLRW space-time,

we do have a set of two diﬀerential equations in which the ﬁrst one determines acceleration whereas the other one

describes the rate of expansion (Hubble parameter) which involves the density of matter. We have statistically

estimated the present values of model parameters, EoS (ω0), the Hubble (H0), decelerating parameters (q0), and

λ. For this, we consider three types of observational data sets: the 46 Hubble parameter data set, the SNe Ia 715

data sets of distance modulus and apparent magnitude, and the 66 Pantheon data set (the latest compilation of SN

Ia 40 bined plus 26 high red shift apparent magnitude mbdata set in the red shift ranges from 0.014 ≤z≤2.26).

These sets of data are compared with the theoretical results through the χ2statistical test and estimated values

are obtained on the basis of minimum χ2. The model exhibits a phase transition from deceleration to acceleration.

We have calculated transitional red shifts and time for the data sets. Our estimated results for the present values

of various model parameters such as the Hubble, deceleration, etc., are found as per expectations and surveys.

The higher value of the present density is attributed to the presence of additional energies in the universe apart

from baryon energy. We have also examined the behavior of the pressure in our model. It is negative and produces

an acceleration in the universe. Its present value is obtained as p0≃ −0.7ρ0.

The outline of the paper is as follows: In section II, the f(R, T ) gravity ﬁeld equations along with the action

and the three speciﬁc functional forms of f(R, T ) are described. In sec. III, the f(R, T ) ﬁeld equations are obtained

for the linear form of f(R, T ) = R+ 2λT in the framework of the FLRW spatially ﬂat space-time. In this section,

we have solved the ﬁeld equations to ﬁnd the expressions for the Hubble and deceleration parameters. In section

IV, the distance modulus, luminosity distance, and apparent magnitude are deﬁned and formulated. Statistical

estimation and evaluation of the model parameters are done in sections V and VI. In these sections, we have

plotted various error bars, and likelihood graphs, the 1σand 2σconﬁdence regions and the deceleration parameter

(q), jerk parameter (j) and snap parameter (s) versus red shift (z) graphs. We have obtained transitional red shifts

and corresponding times which display how the universe passed from the deceleration to the acceleration era. In

section VII, a state ﬁnder analysis is carried out which tells us that our model at present is in quintessence and

its evolution passed through the Einstein - de Sitter and ΛCDM stages. In the last section, we have summarized

the work with the conclusion.

2 f(R,T) gravity

The Einstein ﬁeld equations (EFE) are given by:

Rij −1

2Rgij + Λgij =8πG

c4Tij ,(1)

where the symbols have their usual meanings. These Eqns. are obtained from the following action:

S=Z(1

16πG(R+ 2λ) + Lm)√−gdx4.(2)

Harko et al. [44] modiﬁed the GRT ﬁeld equations by replacing the Ricci scalar R with an arbitrary function

f(R, T ) of Rand the trace Tof the energy-momentum tensor Tij . The action for f(R, T ) gravity is:

S=Z1

16πGf(R, T ) + Lm√−gdx4,(3)

where Lmdenotes the matter Lagrangian density. The stress-energy tensor of the matter is deﬁned as [43]:

Tij =−2

√−g

δ(√−gLm)

δgij (4)

By taking the variation of the action Swith respect to the metric tensor components gij , the ﬁeld equations of

f(R, T ) gravity are obtained as [44]:

Rij −1

2Rgij =8πGTij

fR(R, T )+1

fR(R, T )1

2gij (f(R, T )−Rf R(R, T ))−(gij 2−∇i∇j)fR(R, T )+fT(R, T )(Tij +pgij ),

(5)

where fRand fTdenote the derivatives of f(R, T ) with respect to Rand T, respectively. The Lagrangian for a

perfect ﬂuid is Lm=−p, and its energy momentum tensor is:

Tij = (ρ+p)uiuj−pgij ,(6)

where ρand pare the energy density and pressure, respectively. The vector ui= (0,0,0,1) is the four-velocity

in the co-moving coordinate system which satisﬁes the conditions uiui= 1 and uiui;j= 0. In [44], the authors

proposed the following three cases for the function f(R, T ) for cosmological applications:

f(R, T ) = 





R+ 2f(T)

f1(R) + f2(T)

f1(R) + f2(R)f3(T)





Numerous authors [45, 46, 47, 48, 49] have recently examined in detail the cosmological implications for the class

f(R, T ) = R+ 2f(T). According to Fisher and Carlson’s recent study of f(R, T ) gravity [50], the term f2(T)

should be included in the matter Lagrangian Lm, and hence has no physical meaning. They concentrated especially

on the scenario where fis separable, resulting in f(R, T ) = f1(R) + f2(T). Harko and Moraes [51] thoroughly

reexamined the ﬁndings of the paper [52] and demonstrated that their physical analyses and interpretation of

the T-dependence of f(R, T ) gravity contained signiﬁcant conceptual problems. We refer to recent publications

[53, 54, 55, 56, 57, 59, 60] for a better grasp of the cosmological implications and mathematical structure of

f(R, T ) gravity. We plan to investigate a cosmological model based on f(R, T ) theory which ﬁts best with current

observations, and which can be compared with the ﬁndings of the ΛCDM model. So we take the simple linear

case f(R, T ) = R+λT where λis a scalar that couples Rand T.

3 Metric and Field Equations

The FLRW spatially ﬂat space-time is given as:

ds2=dt2−a2(t)(dx2+dy2+dz2),(7)

where a(t) represents the scale factor. The trace of the stress energy-momentum tensor is obtained as:

T=ρ−3p(8)

The f(R, T ) ﬁeld equations (5) for the metric (7) are obtained as:

2˙

H+ 3H2=−(8π+ 3λ)p+λρ (9)

and

3H2= (8π+ 3λ)ρ−λp, (10)

where the Hubble parameter, H=˙a

a. We assume λ= 8πη. Then the ﬁeld equations (9) and (10) are written as:

(1 −2q)H2= 8π(ηρ −(1 + 3η)p) (11)

and

3H2= 8π((1 + 3η)ρ−ηp),(12)

where q=−¨a

aH2is deceleration parameter.

As the universe is currently accelerating, so both the deceleration parameter and pressure must be negative.

The observations tell us that the luminous content of the universe (baryon ﬂuid) is dust at present so the baryon

pressure must be zero. But the literature [30] says that apart from baryon matter, other forms of matter do exist

in the universe. It is estimated that nearly 28% and 68% of the total content of the universe is the dark matter

and dark energy respectively. The dark matter is responsible for the phenomenon of gravitational lensing and

dark energy is for the present-day acceleration in the universe. These ideas and how to accommodate them in

theories have been explained in the introduction. In f(R, T ) gravity, the Ricci scalar Ris replaced by an arbitrary

function of Rand T. The idea behind is that to have acceleration due to curvature and trace dominance. The

authors feel that the pressure term arising in the ﬁeld equations is not due to the baryon matter but a result of

the overall eﬀect. We mean that terms containing ηin the ﬁeld equations (11) and (12) are the extra terms in

the original FLRW ﬁeld equations of general relativity, and they will have an impact in producing pressure and

creating acceleration in the universe. We observe that there are two Eqs. (11) and (12) with four unknowns,

viz., H,q,pand ρ. Therefore, to get an explicit solution to the above equations, we need to assume at least

one reasonable relation among the variables or we may parameterize the variables. For this, we assume the usual

equation of state for the ﬂuid as p=ω ρ, and we consider the parameterization of the equation of state parameter

(ω) as given by Gong and Zhang[61]:

ω=ω0

(1 + z),

where zis the red shift and ω0is the present value of ω.

Eq. (12) is re-written in the following form:

8πρ0

3H2

1 + k, k =η(3 −ω0), ω0= 3 −k

η(13)

where the suﬃx 0 denotes the present values of the parameters. From this, we can ﬁnd

ρ0=3H2

8π(1 + k)=ρc

1 + k,

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Anf(R,T)GravityBasedFLRWModelandObservationalConstraintsAnirudhPradhan1,GopikantGoswami2,RitaRani3,AroonkumarBeesham4,51CentreforCosmology,AstrophysicsandSpaceScience(CCASS),GLAUniversity,Mathura-281406,UttarPradesh,India2,3DepartmentofMathematics,NetajiSubhasUniversityofTechnology,Delhi,India4Depar...

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An fRT Gravity Based FLRW Model and Observational Constraints Anirudh Pradhan1 Gopikant Goswami2 Rita Rani3 Aroonkumar Beesham45 1Centre for Cosmology Astrophysics and Space Science CCASS GLA University Mathura-281 406 Uttar

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