Bouncing cosmological models in a functional form of FRgravity A. S. Agrawal 1S. Mishra2 S.K. Tripathy 2 and B. Mishra1 1Department of Mathematics Birla Institute of Technology and Science-Pilani Hyderabad Campus Hyderabad-500078 India.

2025-04-30 0 0 680.89KB 13 页 10玖币

Bouncing cosmological models in a functional form of F(R)gravity

A. S. Agrawal ,1, ∗S. Mishra,2, †S.K. Tripathy ,2, ‡and B. Mishra 1, §

1Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India.

2Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India.

Abstract We have investigated some bouncing cosmological models in an isotropic and homo-

geneous space time with the F(R)theory of gravity. Two functional forms of F(R)have been

investigated with a bouncing scale factor. The dynamical parameters are derived and analysed along

with the cosmographic parameters. The analysis in both the models show the occurrence of bouncing

scenario. The violation of strong energy conditions in both models is also shown. In the stability

point of view we have analysed the behaviour of FR=dF

dR with respect to cosmic time and both the

models exhibit stable behaviour.

Keywords:F(R)gravity, Perfect ﬂuid, Bouncing cosmology, Cosmographic parameters, Energy conditions.

I. INTRODUCTION

The initial singularity is another important issue that General Relativity (GR) has encountered among other issues

during early Universe. Friedmann [1,2] claimed that the occurrence of initial singularity was during the beginning

of the evolution of Universe. It is believed that singularity issue occurred before the inﬂation, because the inﬂation-

ary scenario resolved certain key issues of early Universe [3–5]. One possible solution might be the Universe does

not attained singularity during the contraction, but expands after experiencing a bounce. This concept is known

as the big bounce. Recent discoveries [6–12], have revealed that our Universe is undergoing a late time accelerated

expansion phase, which is explained by dark energy, time-independent vacuum energy (according to the ΛCDM

model). The cosmological constant [13], scalar ﬁelds (including quintessence, phantom, quintom, tachyon, and

others) [14–19], and holographic models [20] are possibilities for describing dark energy scenarios. Modiﬁed gravity

theory has advantages over other models since it avoids expensive numerical computations and is consistent with

current data for a late phase accelerating Universe and dark energy. So the models with such theories are being

designed to modify the standard nature of GR by replacing the Ricci scalar Rin Einstein-Hilbert action with f(R).

Several modiﬁed theories of gravity have been developed, such as f(R)gravity [21–29], f(G)gravity [30], f(T)

gravity [31,32] and f(R,T)gravity [33–43], Teleparallel gravity [44,45], where Tdenotes the torsion scalar and Gis

the Gauss Bonnet invariant term. Some other important work on modiﬁed theories of gravity [46–49] are available

in the literature. Most recent f(Q)gravity or symmetric teleparallel gravity [50] and f(Q,T)[51] gravity have been

proposed, where Qand Trespectively represent the non-metricity and trace of energy momentum tensor.

The inﬂationary scenario has been challenged, and the matter bounce scenario has been presented as a possible al-

ternative to address the initial singularity issue. The Universe goes through an initial matter dominated contraction

phase, then a non singular bounce, and ﬁnally a causal generation for ﬂuctuation in the bouncing scenario. For this,

the bouncing scenario is a typical example, and a null energy condition (NEC) has to be violated to realize a solution

in a spatially ﬂat FLRW metric in GR. The matter bounce scenario has gained a lot of attention among the numerous

bouncing models proposed because it creates a scale-invariant power spectrum. Additionally, the Universe passes

through a matter-dominated epoch at the late time in a matter bounce scenario. Alternative gravity theories like

f(R)gravity [52–55], f(G)gravity [56,57], f(R,T)gravity [58–62], f(Q,T)gravity [63], f(T)gravity [64], f(Q)

gravity [65] and f(R,G)gravity [66]have all successfully studied bouncing cosmologies. The present work is on

the bouncing model in a modiﬁed theory of gravity, the f(R)theory in an FLRW space -time. To note, f(R)gravity

theory is an excellent alternative to the standard gravity model to study the dark energy cosmological models. In

∗agrawalamar61@gmail.com

†sachidanandamishra1998@gmail.com

‡tripathy sunil@rediffmail.com

§bivu@hyderabad.bits-pilani.ac.in

arXiv:2210.09726v2 [gr-qc] 12 Apr 2023

the f(R)modiﬁed gravity framework, Odintsov and Oikonomou [67] have investigated a bouncing cosmology with

a Type IV singularity at the bouncing point. Elizalde et al. [68] et al. have studied the extended matter bounce

scenario in ghost-free f(R,G)gravity which is compatible with the gravitational waves.

f(R)gravity is an important and well-known modiﬁed gravity theory [69,70]. Hu and Sawicki proposed a late-

time cosmological model [71], the model can explain the late-time acceleration of the Universe, without the need

of dark energy. Starobinsky [72] suggested a f(R)gravity model that is in line with cosmological conditions and

accords with laboratory experiments and observations of the solar system. However, the exponential gravity model

was developed and studied by Cognola et al. [73]. This model accurately captures the natural inﬂation of the

early Universe and the accelerated expansion of the present Universe. Further these models are studied to get the

traversable wormhole solutions gravity by Shamir and Fayyaz [74]. Chen et al. [75] studied the matter power spectra

with the dynamical background evolution in f(R)theory.

In the context of f(R)gravity, Odintsov et al. [76] have proposed a cosmological model that merges a non-singular

bounce to a matter-dominated epoch and space-time dominated to a late time accelerating epoch; i.e., the model is

similar to a generalized matter bounce model which is also compatible with the late DE dominant phase of the cosmic

evolution. Odintsov et al. [77] investigated a Chern-Simons corrected f(R)gravity theory of a non-singular bounce

to a dark energy epoch, where the Chern-Simons coupling function is supposed to have a power law behaviour

with the Ricci scalar. In a gravitational model with curvature-squared R2and curvature quartic R4non-linearities,

Saidov and Zhuk [78] have examined bouncing inﬂation. Barragan et al. [79], have analysed the criteria that guar-

antee the existence of homogeneous and isotropic models that avoids the Big Bang singularity in Palatini formalism.

The modiﬁed Friedmann equation in LQC has been transferred to the Palatini f(R)theory [80] by Olmo and Singh

whereas Olmo [81], has discovered the necessary f(R)function that must be taken into account to create a bouncing

cosmology of this type of LQC. In the generalized f(R)theory, Nojiri et al. [82] have studied non-singular bounce

cosmology in the context of Lagrange multiplier. As a result, it is discovered that the weak energy and null en-

ergy conditions are violates close to the bouncing point. A common f(R)gravity model is used to describe the

phenomenology of the current non-singular bounce.

In this paper, our objective is to study some bouncing cosmological models to avoid the initial singularity issue

with some of the functional forms of F(R) = R+f(R),f(R)is the deviation of F(R)from the Einstein gravity.

To explain the late-time cosmic speed-up issue, the models will look at geometrical degrees of freedom. The ex-

planation of F(R)gravity and the derivation of F(R)ﬁeld equations are presented in Sec. II of the study. In Sec.

III, the bouncing scale factor and Hubble parameter were introduced. Two models with the bouncing scale factor

and functional form of F(R)are provided in Sec. IV. The cosmographic parameters are discussed in Sec. V and the

energy conditions of both models are given in Sec. VI. Stability analysis has been done in Sec. VII. The model results

and conclusions are presented in Sec.VIII.

II. FIELD EQUATIONS OF F(R)GRAVITY

The action for F(R)gravity can be deﬁned as,

S=Zp−gF(R)

2κ2d4x, (1)

κ2=8πG

c4,Gbe the Newton’s gravitational constant, gis determinant of the metric tensor gij. Varying action (1)

with respect to gij, the F(R)gravity ﬁeld equations can be obtained as,

FRRij −1

2Fgij − ∇i∇jFR+gijFR=0 (2)

Here FR=dF

dR ,∇irepresents the covariant derivative, ≡gij∇i∇jis the d’Alembert operator. The natural system

of unit 8πG=¯

h=c=1 has been used, where G,¯

hand crespectively denote the Newtonian gravitational constant,

reduced Planck constant and velocity of light in vacuum respectively.

We consider the ﬂat FLRW space-time as,

ds2=−dt2+a2(t)(dx2+dy2+dz2), (3)

For this metric, the temporal and spatial components of the Eq. (2) becomes

0=−F

2+3H2+˙

HFR−18 4H2˙

H+H¨

HFRR (4)

0=F

2−3H2+˙

HFR+68H2˙

H+4˙

H2+6H¨

H+˙

HFRR +36 4H˙

H+¨

H2FRRR (5)

where FRR =d2F

dR2,FRRR =d3F

dR3and H=˙

ais the Hubble parameter. When the above equations are compared to

the standard Friedmann equations, it is clear that F(R)gravity contributes to the energy-momentum tensor, with its

effective energy density ρe f f and pressure pe f f given by

ρeff =−f

2+3H2+˙

HfR−18 4H2˙

H+H¨

HfRR (6)

peff =f

2−3H2+˙

HfR+68H2˙

H+4˙

H2+6H¨

H+˙

HfRR +36 4H˙

H+¨

H2fRRR (7)

Eqns. (6) and (7) can be expressed in terms of Hubble parameter, H=˙

aand the derivatives of functional form of

F(R)with respect to Rin which the Ricci scalar, R=6¨

a+˙

a2. So, we need a Hubble function to obtain the energy

density and pressure of the matter ﬁeld to further study the dynamics of the Universe. Also to study the issue of late

time acceleration issue, the equation of state (EoS) parameter behaviour to be analysed, which can be obtained as,

ωeff =peff

ρeff

=−1+12 2H2˙

H+4˙

H2+3H¨

H+˙

HfRR +72 4H˙

H+¨

H2fRRR

f−6H2+˙

HfR+36 4H2˙

H+H¨

HfRR

(8)

Where f(R)represents the departure of F(R)gravity from Einstein gravity, F(R) = R+f(R). As expected, the

effective energy-momentum tensor relies on the nature of F(R). So, in the subsequent sections, we will study the

bouncing scenario and late time cosmic acceleration issue of the Universe by considering the bouncing scale factor

and some of the functional forms of F(R).

III. THE SCALE FACTOR

Inﬂationary cosmology is one of two extant theories of the early Universe, with the other being bounce cosmology,

in which the theoretical contradictions of the Big Bang description of our Universe are addressed. The most recent

observational data imposed strict limits on inﬂationary models, conﬁrming the validity of some while ruling out

others. Here we intend to study the bouncing scenario in the F(R)theory of gravity.

• For the case of non-singular bounce, bouncing scenario behaves as a contracting nature formulated by the

scale factor which decreases with time, i.e., ˙

a<0, means Hubble parameter is negative in contacting phase,

i.e., H=˙

a/a<0.

• For bouncing epoch, contracting nature of scale factor to a non zero ﬁnite critical size is obtained as a result of

which the Hubble parameter vanishes at bounce making H=0.

• Nature of scale factor increases with time in a positive acceleration, so as the Hubble parameter becomes

positive after the bounce, i.e., ˙

a>0.

• For the situation of near to bouncing epoch, Hubble parameter holds true, i.e., ˙

H>0 which is suitable for

ghost (phantom) behaviour of the model.

• Also to appreciate a bouncing model, EoS evolves at such phantom region and changes twice, one before the

bounce and another after the bounce.

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BouncingcosmologicalmodelsinafunctionalformofF(R)gravityA.S.Agrawal,1,S.Mishra,2,S.K.Tripathy,2,andB.Mishra1,§1DepartmentofMathematics,BirlaInstituteofTechnologyandScience-Pilani,HyderabadCampus,Hyderabad-500078,India.2DepartmentofPhysics,IndiraGandhiInstituteofTechnology,Sarang,Dhenkanal,Odisha-...

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Bouncing cosmological models in a functional form of FRgravity A. S. Agrawal 1S. Mishra2 S.K. Tripathy 2 and B. Mishra1 1Department of Mathematics Birla Institute of Technology and Science-Pilani Hyderabad Campus Hyderabad-500078 India.

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