Advances in High Energy Physics

Volume 2017 (2017), Article ID 6378904, 10 pages

https://doi.org/10.1155/2017/6378904

## Anisotropic Universe in Gravity

Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore, Pakistan

Correspondence should be addressed to M. Farasat Shamir

Received 14 July 2017; Revised 2 November 2017; Accepted 14 November 2017; Published 12 December 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 M. Farasat Shamir. 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 investigating the recently introduced theory of gravity, where is the Gauss-Bonnet term and is the trace of the energy-momentum tensor. For this purpose, anisotropic background is chosen and a power law gravity model is used to find the exact solutions of field equations. In particular, a general solution is obtained which is further used to reconstruct some important solutions in cosmological contexts. The physical quantities like energy density, pressure, and equation of state parameter are calculated. A Starobinsky-like model is proposed which is used to analyze the behavior of universe for different values of equation of state parameter. It is concluded that presence of term in the bivariate function may give many cosmologically important solutions of the field equations.

#### 1. Introduction

Some important modifications of general relativity (GR) have been proposed in the last two decades. The mostly discussed theories are and theories of gravity ( is the Ricci scalar and is the trace of energy-momentum tensor) which have been treated most seriously [1–14]. However, recently cosmology has been severely challenged [15]. On the other hand, modified Gauss-Bonnet (GB) gravity is another theory which has gained popularity in the last few years [16–18]. It is also known as theory of gravity, where is a generic function of GB invariant . GB term plays an important role as it may avoid ghost contributions and helps in regularizing the gravitational action [19]. Thus, to save theories, one may include terms. A theory with a similar idea has been recently proposed named as gravity [20]. Some interesting work has been done in the recent past using modified GB theories.

Anisotropic compact stars in modified gravity have been discussed by Abbas et al. [21]. Houndjo et al. [22] found the exact solutions of field equations using cylindrical symmetry and it was concluded that there existed seven families of exact solutions for three different forms of models. The exact cylindrically symmetric solutions of modified field equations recovered cosmic string space-time [23]. A more generalized version of GB gravity known as gravity has also been discussed widely. Wu and Ma [24] investigated the spherically symmetric solutions at low energy where the weak-field and slow-motion limit of gravity was developed. Laurentis et al. [25] discussed cosmological inflation in theory. Sharif and Ikram [26] examined warm inflation in theory of gravity using scalar fields for the Friedmann-Robertson-Walker (FRW) universe model. Conserved quantities have been recently explored in FRW background using Noether symmetry [27]. Sharif and Fatima [28] discussed the role of GB term for the early and late-time accelerating phases of the universe by considering two viable models. García et al. [29] explored energy conditions to prove the viability of some gravity models. gravity energy conditions have been recently explored where the WEC was used along with the recent estimated values of cosmological parameters to determine the viability of some specific choices of gravity models [30].

In the context of gravity, Sharif and Ikram proved that the massive test particles followed nongeodesic lines of geometry due to the presence of extra force and examined the energy conditions for flat FRW universe [20]. The same authors [31] used reconstruction techniques to reproduce the cosmic evolution corresponding to de Sitter universe, power law solutions, and phantom/nonphantom eras in this theory. In a recent paper [32], theory of gravity was discussed using Noether symmetry approach. Two specific models were studied to determine the conserved quantities and it was concluded that the well-known de Sitter solution could be reconstructed for some specific choice of gravity model. In a recent paper, Sharif and Ikram [33] studied the wormhole solutions using power law gravity models and it was shown that traversable wormhole solutions were physically acceptable in theory of gravity. In another work [34], Noether symmetry methodology has been used to study some cosmologically important gravity models with anisotropic background. It is concluded that the specific models of modified GB gravity may be used to reconstruct CDM cosmology without involving any cosmological constant. For some particular choices of gravity models, it is anticipated that this theory may explain the late-time cosmic acceleration. Thus it seems interesting to explore further the modified gravity. Moreover, in comparison with gravity, the presence of the matter term in the bivariate function may give many solutions of the field equations and the theory may support the accelerated expansion of universe under certain conditions for the model under consideration.

In this paper, we are focussed to investigate the dynamics of gravity with anisotropic background. It is well-known that the isotropic models are among the best choices to study large scale structure of the universe. Moreover, according to the cosmological observations including the cosmic microwave background (CMB) radiation, the current universe is isotropic. However, it is believed that the early universe may not have been exactly uniform. Also the local anisotropies that we observe today in galaxies and super clusters also motivate us to model the universe with anisotropic background. Bianchi type models are among the simplest models with anisotropic background. In particular, the investigation of Bianchi type universe in context of modified theories is interesting. In this work, we are interested to explore gravity using locally rotationally symmetric (LRS) Bianchi type space-time. We find the exact solutions of the LRS Bianchi type field equations in theory of gravity. In particular, a general solution with power law gravity model is reported. The plan of the paper is as follows: some basics of gravity and field equations are discussed in Section 2. Section 3 is devoted to exploring the exact solutions of modified field equations. Section 4 is used to reconstruct some important cosmological solutions. Final remarks are given in the last section.

#### 2. Some Basics of Gravity

The modified GB gravity is given by the action [20]

Here and denote the GB term and the trace of the energy-momentum tensor, respectively, whereas is the standard matter Lagrangian, is the Ricci scalar, is the determinant of metric tensor, and is a coupling constant. The field equations can be obtained by varying the action equation (1) with respect to the metric tensor [20]where the symbols involved have their usual meanings and and the subscript or in the functions denotes the partial derivatives. It would be interesting to notice that if we substitute in (2), then the field equations of gravity are recovered. Moreover, the case reduces the modified field equations to the usual GR equations. For the sake of simplicity, from now onwards we consider , , and so forth. The trace of (2) gives

It may be noticed that this relates , , and differentially and not algebraically as in GR, where . This indicates that the modified field equations may admit many solutions in addition to other modified theories and GR. The covariant divergence of (2) is given bywhich is not zero. It is due to the presence of higher order derivatives of the energy-momentum tensor that are naturally present in the field equations. Thus the theory might be plagued by divergences at astrophysical scales. This seems to be an issue with some other higher order derivatives theories as well that include higher order terms of energy-momentum tensor. However, to deal with the issue, one can put some constraints to (4) to obtain standard conservation equation [20]. Here we take the spatially homogeneous, anisotropic, LRS Bianchi type space-timewhere and are cosmic scale factors. Moreover, we consider that the universe is composed of perfect fluidwhere and denote the energy density and pressure of the fluid, respectively. The Ricci scalar and GB invariant for (5) are given aswhere the overdot denotes the derivative with respect to the time coordinate. Now we define some textbook physical quantities. The average scale factor and average Hubble parameter for the model under consideration take the form

The expansion scalar and shear scalar are given as follows:whereand is the projection tensor. Now, for LRS Bianchi type space-time (5), the field equations (2) take the form

These are three highly nonlinear and difficult differential equations with five unknowns. Thus we need an additional constraint to investigate any exact solution. Here we may consider a physical condition that shear scalar is proportional to expansion scalar which provideswhere is an arbitrary real number. In literature [35–38], many authors explored the exact solutions of field equations using this condition. Thus, using (12), field equations (11) take the form

Now we investigate the exact solutions of these field equations.

#### 3. Exact Solutions of Modified Field Equations

We consider the model aswhere and are arbitrary constants. Further, we choose in power law form; that is,

This model has already been proposed by Cognola et al. [18] and it is interesting because the chances of the appearing of Big-Rip singularity vanish using this model. Subtraction of (14) and (15) yields

Using (17) in (16), it follows that

For simplicity and without loss of any generality, we choose so that (18) takes the form

After inserting the value of GB invariant (20) reduces to differential equations with three unknowns , , and . It would be worthwhile to mention here that many solutions can be found using (20). Here we consider the power law form; that is,where and are arbitrary constants. Using this in (20), we obtain a constraint equation

This equation is satisfied for such that

Thus, corresponding to two roots of this equation, we obtain two choices of models:where and are integration constants. The first model recovers the usual GB gravity for and . The second model with square root term is important as it leads to a viable inflation in the presence of massive scalar field [39]. It is mentioned that different forms of can be assumed to reconstruct the solutions. However, we propose only two models for the present analysis.

*Case I (linear model). *We consider linear form here for the sake of simplicity. Thus, considering and using (13)–(15), the expressions for energy density and pressure of universe turn out to be

It is evident from (25) that energy density and pressure of the universe are defined for and for . Adding (25), we obtain

The behaviour of energy density plus pressure of universe can be seen from Figure 1(a) for the model with . It is clear that as time grows which further suggests EoS parameter . This is interesting as the phantom-like dark energy is found to be in the region where . The universe with phantom dark energy ends up with a finite time future singularity known as cosmic doomsday or Big Rip [40, 41]. Moreover, accelerated expansion of universe is described when [42–44]. Figure 1(b) depicts the behavior of for radiation universe.