Advances in High Energy Physics

Volume 2018, Article ID 9534279, 11 pages

https://doi.org/10.1155/2018/9534279

## Study of Homogeneous and Isotropic Universe in Gravity

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan

Correspondence should be addressed to M. Sharif; kp.ude.up@htam.firahsm

Received 7 February 2018; Revised 25 May 2018; Accepted 26 June 2018; Published 26 July 2018

Academic Editor: Ricardo G. Felipe

Copyright © 2018 M. Sharif and Aisha Siddiqa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

This paper is devoted to study the cosmological behavior of homogeneous and isotropic universe model in the context of gravity, where is the scalar field. For this purpose, we follow the first-order formalism defined by . We evaluate Hubble parameter, effective equation of state parameter , deceleration parameter, and potential of scalar field for three different values of . We obtain phantom era in some cases for the early times. It is found that exponential expression of yields independent of time for flat universe and independent of model parameter otherwise. It is concluded that our model corresponds to CDM for both initial and late times.

#### 1. Introduction

The most striking and fascinating area in cosmology is the current accelerated expansion of the universe suggested by various observations. The agent causing this expansion is termed as dark energy and it violates the strong energy condition. Cosmological observations reveal that our universe is approximately homogeneous and isotropic at large scales [1] described by the standard FRW model. Inclusion of cosmological constant () to the standard model leads to cold dark matter (CDM) model. In general relativity (GR), the CDM model explains the expanding behavior of the universe where is supposed to play the role of dark energy.

Despite its many features, there are two major issues associated with this model named as fine tuning and coincidence problems [2]. The huge difference between the vacuum energy density and the ground state energy suggested by quantum field theory leads to the first issue. For this model, the densities of dark matter and dark energy are of the same order leading to cosmological coincidence problem. In order to resolve such and other open issues, alternative models of dark energy were proposed by modifying either matter or geometric part of the Einstein-Hilbert action. Modified matter models [3–7] are quintessence, phantom, K-essence, holographic dark energy, and Chaplygin gas. Some examples of modification in geometric part are scalar-tensor theory, and theories of gravity.

Harko* et al*. [8] proposed gravity as a generalized modified theory where is the Ricci scalar and is the trace of energy-momentum tensor. The dependence on is included due to the considerations of exotic fluids or quantum effects. The coupling of curvature and matter yields interesting consequences such as covariant derivative of the energy-momentum tensor which is no longer zero implying the existence of an extra force as well as nongeodesic path of particles. In cosmological scenario, it can explain the problem of galactic flat rotation curves as well as dark matter and dark energy interactions [9]. Jamil* et al*. [10] introduced some cosmic models in this gravity and showed that dust fluid can reproduce CDM model. Sharif and Zubair [11] explored thermodynamics and concluded that generalized second law of thermodynamics is valid for phantom as well as nonphantom phases. The same authors [12] established energy conditions and constraints for the stability of power-law models.

Singh and Singh [13] discussed the reconstruction of gravity models in the presence of perfect fluid and showed that extra terms can represent phantom dark energy as well as cosmological constant in the presence and absence of perfect fluid, respectively. Singh and Kumar [14] explored the role of bulk viscosity using FRW model with perfect fluid and found that bulk viscosity provides a supplement for expansion. In literature [15–17], the higher dimensions are also explored in the framework of gravity. Moraes* et al.* [18] studied hydrostatic equilibrium condition for neutron stars with a specific form of equation of state (EoS) and found that the extreme mass can cross observational limits.

In [8], Harko* et al.* also discussed theory where is the trace of energy-momentum tensor of a scalar field. Scalar fields have extensively been studied in cosmology during the last three decades. Yukawa is the pioneer to introduce scalar field in physics which naturally has an EoS and hence is expected to play a vital role in modern cosmology as well as astrophysics [19]. Alves* et al.* [20] confirmed the existence of gravitational waves for extra polarization modes in both as well as theories.

Halliwell [21] explored the role of scalar field with exponential potential and also discussed some examples that yield exponential potential. Virbhadra* et al.* [22] studied the effect of scalar field in gravitational lensing and explored the features of lens specified by mass and charge of scalar field. Nunes and Mimoso [23] worked on phase-plane analysis for a flat FRW model in the presence of perfect fluid as well as self-interacting scalar field. They showed that the scalar field potential having positive, monotonic, and asymptotically exponential behavior leads to global attractor. Das and Banerjee [24] explored that the scalar field can produce deceleration and acceleration phases of cosmos by considering an energy transfer between scalar field and dark matter.

Bazeia* et al.* [25] introduced the first-order formalism for scalar field models and also discussed some examples of cosmological interest. In [26], the authors extended this formalism for scalar fields with tachyonic dynamics. Recently, Moraes and Santos [27] presented a cosmological picture in gravity in the absence of matter by considering flat FRW model. They started with the relation which comes from the field equations of GR via the above-mentioned formalism. We have applied this formalism to Bianchi type-I universe model (with no matter present) and concluded that the exponential form of can explain all the stages of the universe evolution [28].

The motivation for neglecting matter action is that framework cannot describe the radiation-dominated era because in that phase the value of for perfect fluid is zero and theory is reduced to scenario [27]. Moreover, Alves* et al.* [20] showed that for gravitational wave passing through vacuum with no extra polarization modes are induced due to curvature matter coupling as in vacuum. On the other hand, the same phenomenon in is controlled by the assumed model. Hence, the study of radiation phase as well as any phenomenon in vacuum has become a crucial issue to be addressed. In this regard, Moraes introduced an extra dimension [29] and in [30] he considered a varying speed of light to obtain the solutions in radiation phase.

In this paper, we explore cosmology of generalized FRW universe model using the same action used in [27] and with the fundamental assumption of first order formalism, i.e., in gravity. The plan of the paper is as follows. In the next section, we formulate the field equations and find the values of , , , and . We also discuss graphical behavior of these parameters for different models of . The last section provides the obtained results.

#### 2. Field Equations and First-Order Formalism

The action for gravity is given bywhere we assume that . In the following, we use the model [27], where is a constant known as model parameter. The corresponding field equations arewhere is the Einstein tensor and represents the energy-momentum tensor of a scalar field. For a real , the Lagrangian density and the energy-momentum tensor are given bywhere denotes the self-interacting potential. The trace of the energy-momentum tensor isdot denotes derivative with respect to .

The line element of FRW model is given bywhere is the scale factor and represents curvature of the space. For , we have flat, closed, and open universe model, respectively. The corresponding field equations arewhich yield the expression of as The equation of motion for scalar field is obtained aswhere subscript indicates derivative with respect to . Following the first-order formalism [25], the Hubble parameter is given byThe expressions of and from the field equations becomewhere and are while the decelerating parameter is defined asSubstituting from (11) into (9), we have which is a quadratic equation in . It has two rootseach of which is a first-order differential equation. We find the solution for the following three expressions of .(1), where is a real constant. The scalar field potentials followed by this value of are of much interest. When , it reproduces some negative potential and leads to potential which presents spontaneous symmetry breaking [25].(2), where is a real constant. It represents a type of model [31].(3), is a real constant. It is a sine-Gordon type of model [32].

##### 2.1. Flat Universe

In this case, (17) implies that eitherFor , we have constant value of and consequently (11)-(15) yield which represents CDM model. The solutions of the second option in (18) for all forms of are given in Table 1, where , and are constants of integration. The graphical behavior of the corresponding , , , and is shown in Figures 1–3. For graphical analysis, we have taken the free parameters such that when , is increasing while if , then is decreasing and when , is constant [33]. According to standard cosmology [34], after inflation and it becomes constant with the passage of time. We see that the Hubble parameter shows this behavior only for exponential value of . The Hubble parameter and are inversely related, i.e., decrease in increases and vice versa for all values of .