Advances in High Energy Physics

Volume 2018, Article ID 2854567, 11 pages

https://doi.org/10.1155/2018/2854567

## Bianchi Type-I Dark Energy Cosmology with Power-Law Relation in Brans-Dicke Theory of Gravitation

^{1}Department of Mathematics, SGBAU, Amravati 444 602, India^{2}Department of Mathematics, PRMIT&R, Badnera, Amravati 444 701, India

Correspondence should be addressed to D. V. Kapse; moc.liamg@espak.itpid

Received 3 February 2018; Revised 17 April 2018; Accepted 4 June 2018; Published 9 July 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 S. D. Katore and D. V. Kapse. 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

We have studied the interacting and non-interacting dark energy and dark matter in the spatially homogenous and anisotropic Bianchi type-I model in the Brans-Dicke theory of gravitation. The field equations have been solved (i) by using power-law relation and (ii) by assuming scale factor in terms of redshift. Here we have considered two cases of an interacting and non-interacting dark energy scenario and obtained general results. It has been found that for suitable choice of interaction between dark energy and dark matter we can avoid the coincidence problem which appears in the model. Some physical aspects and stability of the models are discussed in detail. The statefinder diagnostic pair, i.e., , is adopted to differentiate our dark energy models.

#### 1. Introduction

The recent cosmological observational data of Type Ia Supernovae (SNeIa) (Riess et al. [1]; Perlmutter et al. [2]), Cosmic Microwave Background (CMB) (Bennett et al. [3]; Spergel et al. [4]), Large Scale Structure (LSS) (Tegmark et al. [5, 6]), the Sloan Digital Sky Survey (SDSS) (Seljak et al. [7], Adeleman-McCarthy et al. [8]), Wilkinson Microwave Anisotropy Probe (WMAP) (Knop et al. [9]), and Chandra X-ray observatory (Allen et al. [10]) strongly suggests that our universe is dominated by a component with large negative pressure called dark energy (DE).

The study of DE is possible through its equation of state (EoS) parameter which is not necessarily constant, where is the pressure and is the energy density of DE. The DE candidate which can simply explain the cosmic acceleration is a vacuum energy , which is mathematically equivalent to the cosmological constant (). The other conventional alternatives, which can be described by minimally coupled scalar fields, are quintessence , phantom , and quintom (that can cross from phantom region to quintessence region). From observational results coming from SNe Ia data (Knop et al. [9]) and combination of SNe Ia data with CMBR anisotropy and galaxy clustering statistics (Tegmark et al. [8] ), the limits on EoS parameter are obtained as −1.67 < < −0.62 and −1.33 < < −0.79, respectively. Recently, DE models with variable EoS parameter have been studied by Ram et al. [11, 12], Katore et al. [13], Reddy et al. [14], and Mahanta et al. [15].

Interaction between DE and dark matter (DM) leads to a solution to the coincidence problem (Cimento et al. [16]; Dalal et al*. *[17]; Jamil and Rashid [18, 19]). By considering a coupling between DE and DM, we can explain why the energy densities of DE and DM are nearly equal today. Due to interaction between two components, the energy conservation cannot hold for the individual components. Recent observations (Bertolami et al. [20]; Le Delliou et al. [21]; Berger and Shojaei [22]) provide the evidence for the possibility of such an interaction between DE and DM. Zhang [23, 24], Zimdahl and Pavon [25], Pradhan et al. [26, 27], Saha et al. [28], Amirhashchi et al. [29–33], Adhav et al. [34, 35], and Fayaz [36] have investigated various cosmological models with interacting DE.

The Brans-Dicke theory [37] is a generalized form of general relativity and it is one of the most enchanting examples of scalar tensor theories of gravitation. Brans-Dicke (BD) theory introduces a scalar field which has the dimensions of the inverse of gravitational constant and which interacts equally with all forms of matter. Recently, Rao et al. [38], Sarkar [39, 40], Katore et al. [41], Singh and Dewri [42], and Reddy et al. [43, 44] have studied the cosmological models in Brans-Dicke theory of gravitation.

Amirhashchi et al. [33] have investigated the DE equation of state (EoS) parameter in both interacting and non-interacting cases and examined its future by applying hyperbolic scale factor in general relativity. Motivated by the above investigations, in this paper, we have extended the work of Amirhashchi et al. [33] in Brans-Dicke theory of gravitation. This is relevant because of the fact that scalar field plays an important role in the discussion of DE models. In this paper, we have studied the interacting and non-interacting dark energy and dark matter in the spatially homogenous and anisotropic Bianchi type-I model in the Brans-Dicke theory of gravitation. The field equations have been solved (i) by using power-law relation and (ii) by assuming scale factor in terms of redshift and have discussed the physical properties and also the physical acceptability and stability of our models. The values of cosmological parameters are taken from the recent observations made by Amirhashchi [45], Amirhashchi and Amirhashchi [46, 47], and Patrignani et al. [48]. The paper has following structure. In Section 2, the metric and the Brans-Dicke field equations are described. Section 3 is devoted to the solution of the field equations. Using the scale factor as a function of redshift, we have obtained our results for non-interacting and interacting cases. In Section 4, we have discussed the physical aspects and stability of models. In Section 5, behavior of anisotropy parameter of expansion is studied. The statefinder diagnostic pair, i.e., , is adopted to characterize different phases of the universe in Section 6 and finally, Section 7 contains some concluding remarks.

#### 2. The Metric and BD Field Equations

We consider the homogeneous and anisotropic Bianchi type-I universe as where the scale factors , and are functions of time only.

BD field equations for the combined scalar and tensor fields with are given by (Brans-Dicke [37], Reddy et al. [43], and Rao et al. [49]) where is the Ricci scalar, is the Ricci tensor, is the Brans-Dicke scalar field, is the dimensionless constant, and is the energy momentum tensor. The scalar fields satisfy the following equation: The energy momentum tensor is given by where and are energy momentum tensors of DM and DE, respectively. These are given byandwhere and are energy densities of DM and DE, respectively. Similarly, and are the pressure of DM and DE, respectively, while and are the corresponding EoS parameters of DM and DE (Harko et al. [50]).

In comoving coordinate system, the BD field equations (2) and (3) for the metric (1) are given byand the wave equation iswhere an overhead dot denotes differentiation with respect to .

#### 3. Solutions of Field Equations

We have initially six variables and four linearly independent equations (9)-(12). The system is thus initially undetermined and we need additional condition to solve the system completely. In order to solve these field equations, we first assume the power-law relation between the average scale factor () and scalar field () (Pimental [51], Johri and Desikan [52]) aswhere and are constants.

To examine the general results, we assume that the average scale factor is a hyperbolic function of time as Amirhashchi [53]which gives dynamical deceleration parameter (). Also, Chen and Kao [54] have shown that this scale factor is stable under metric perturbation.

In terms of redshift the above scale factor is given bywhere is the redshift parameter and is the average scale factor.

From (10)-(12), we obtainSolving (17)-(19), we obtainwhere , , and are constants of integration.

Equations (20)-(22) further reduce towhere , , and are constants of integration.

Using (23)-(25), we can write the metric functions , and explicitly aswhere , , , , , and , which satisfies the relations and .

Using (26)-(28) in (9), we obtainwhere , , and are the Hubble parameters. For , the model reduces to the flat FRW model in BD theory.

The energy conservation equation is and is given by

##### 3.1. Non-Interacting Dark Energy and Dark Matter

In this section we have considered that there is no interaction between DE and DM. Therefore, the general form of energy conservation equation (30) leads to (Harko et al. [50])andUsing (31), we obtain the energy density of DM aswhere is a constant of integration.

Using (33) and (29), we obtain the energy density of DE aswhere is the energy density of DM and 0 denotes the present value of .

Using (14), (33), and (35) in (10), we obtain the EoS parameter of DE aswhere is the deceleration parameter.

Using (16) in (37), we obtain the EoS parameter of DE in terms of redshift as

The behavior of EoS parameter of DE in terms of redshift is depicted in Figure 1. Here the parameter is taken to be zero and vary constant as 0.01, 0.05, and 0.09, respectively. From figure it is clear that, for all small values of , the EoS of DE is varying in quintessence region and crossing Phantom Divide line (PDL) . However, it is observed that at late time (i.e., at ) the EoS parameter . Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.