Abstract

We have presented FRW cosmological model in the framework of Brans-Dicke theory. This paper deals with a new proposed form of deceleration parameter and cosmological constant . The effect of bulk viscosity is also studied in the presence of modified Chaplygin gas equation of state (). Furthermore, we have discussed the physical behaviours of the models.

1. Introduction

It has been well established that alternative theories of gravitation played an important role in understanding the models of the Universe. For the last few decades, researchers have shown more interest in alternative theories of gravitation especially scalar-tensor theories of gravity. The Brans-Dicke theory (BDT) of gravity is the one of the most successful alternative theories among all alternative theories of gravitation. This theory is consisting of a massless scalar field and a dimensionless constant describing the strength of the coupling between and the matter [1]. In the BDT, gravitational constant is treated as the reciprocal of a massless scalar field , where is expected to satisfy scalar wave equations and its source is all matter in the Universe.

In a pioneering work, both research contributions by Mathiazhagan & Johri [2] and later La & Steinhardt [3] showed that the idea of inflationary expansion with a first-order phase transition can be made to work more satisfactorily if one considers the BDT in place of general relativity. The interesting consequence of BD scalar field is that the modified field equations would express the scale factor as a power function of time and not as an exponential function, so that one attains the so-called “graceful exit” from the inflationary vacuum phase through a first-order phase transition. Hyperextend inflation [4] generalizes the results of extended inflation in BDT and solves the graceful exit problem in a natural way, without recourse to any fine-tuning as required in relativistic models. Romero & Barros [5] discussed the limit of the Brans-Dicke theory of gravity when and showed by examples that, in this limit, it is not always true that BDT reduces to general relativity. From the literature, it is known that the result of BDT is close to Einstein theory of general relativity for large value of the coupling parameter [6, 7]. A more recent bound on the Brans-Dicke parameter is [7]. A number of researchers [8–15] have discussed various aspects of expanding cosmological models in BDT.

Cosmological observations [16, 17] and various related research clearly indicate that the constituent of the present Universe is dominated by dark energy, which constitutes about three-fourths of the whole matter of our Universe. There are several candidates for dark energy like quintessence, phantom, quintom, holographic dark energy, K-essence, Chaplygin gas, and cosmological constant. Among all the dark energy candidates, cosmological constant is the more favoured. It provides enough negative pressure to account for the acceleration and contributes an energy density of same order of magnitude compared to the energy density of the matter [18]. The discrepancy of observed value and theoretical value of cosmological constant is usually referred as cosmological constant problem in literature. This problem is the puzzling problem in standard cosmology. The cosmological constant bears a dynamical decaying character so that it might be large at early epoch and approaching to a small value at the present epoch.

The effect of cosmological constant has been discussed in the literature in the context of general relativity and its alternative theories. T. Singh and T. Singh [19] presented a cosmological model in BDT by considering cosmological constant as a function of scalar field . Exact cosmological solutions in BDT with uniform cosmological “constant” have been studied by Pimentel [20]. A class of flat FRW cosmological models with cosmological “constant” in BDT have also been obtained by Ahmadi-Azar & Riazi [21]. The age of the Universe from a view point of the nucleosynthesis with term in BDT was investigated by Etoh et al. [22]. Azad & Islam [23] extended the idea of T. Singh and T. Singh [19] to study cosmological constant in Bianchi type I modified Brans-Dicke cosmology. Qiang et al. [24] discussed cosmic acceleration in five-dimensional BDT using interacting Higgs and Brans-Dicke fields. Smolyakov [25] investigated a model which provides the necessary value of effective cosmological “constant” at the classical level. Recently, embedding general relativity with varying cosmological term in five-dimensional BDT of gravity in vacuum has been discussed by Reyes & Aguilar [26]. Singh et al. [27] have studied the dynamic cosmological constant in BDT.

On the other side, it is known from the literature that for early evolution of the Universe, bulk viscosity is supposed to play a very important role. The presence of viscosity in the fluid explores many dynamics of the homogeneous cosmological models. The bulk viscosity coefficient determines the magnitude of the viscous stress relative to the expansion. Recently Saadat & Pourhassan [28] investigated the FRW bulk viscous cosmology with modified cosmic Chaplygin gas. Many researchers also have shown interest in FRW bulk viscous cosmological models in different contexts (see Saadat & Pourhassan [28] and references therein).

Motivated by the above studies, here we have discussed the variable cosmological constant for FRW metric in the context of BDT with a special form of deceleration parameter.

2. Field Equations

The field equation of Brans-Dicke theory in presence of cosmological constant may be written aswhere is the scalar field. The energy-momentum tensor of the cosmic fluid in the presence of bulk viscosity may be be defined asLet us consider a homogeneous and isotropic Universe represented by FRW space-time metric aswhere is the curvature parameter, which represents closed, flat, and open model of the Universe and is the scale factor.

The FRW metric (3) and energy-momentum tensor (2) along with Brans-Dicke field equations yield the following equations:

3. Solution of the Field Equations

In order to find exact solutions of basic field equations (4), one must ensure that set of equations should be closed. Thus, two more physically reasonable relations are required among the variables.

First we consider a well accepted power law relation between scale factor and scalar field of the form [27]and as it has been well established the expansion of present Universe is accelerating. In order to study a cosmological model with early deceleration and late time acceleration, we have proposed deceleration parameter of the formas the second physically plausible relation, where . The considered form of deceleration parameter is motivated by the bilinear form of deceleration parameter, Mishra & Chand [29]. Deceleration parameter is useful to classify the models of the Universe. From literature we know that deceleration parameter is a constant quantity or it depends on time. In the case when rate of expansion never changes and is constant, the scaling factor is proportional to time, which leads to zero deceleration. In case when is constant, the deceleration parameter is also constant . In de-Sitter and steady state Universe such cases arises. Now we will classify the cosmological models on the basis of time dependence on Hubble parameter and deceleration parameter as follows, Bolotin et al. [30].(i), : expanding and decelerating;(ii), : expanding and accelerating;(iii), : contracting and decelerating;(iv), : contracting and accelerating;(v), : expanding, zero deceleration/constant expansion;(vi), : contracting, zero deceleration;(vii), : static.From the above classification, (i), (ii), and (v) are possible cases as in the present scenario our Universe is expanding. Again also we have found the following type of expansion exhibit by our Universe.(i): superexponential expansion;(ii): exponential expansion (for known as de-Sitter expansion);(iii): expansion with constant rate;(iv): accelerating power expansion;(v): decelerating expansion.

We consider third physically plausible relation as the modified Chaplygin gas equation of state as follows [31, 32]:where , are constants and .

The set of field equations (4) with the help of (5) may be written asEquations (8) leads us toThis equation is useful for obtaining the various cosmological solutions.

Now our problem is to evaluate the , which is obtained from the relationWith the help of (6) and integrating (10), we obtainedwhere is a constant of integration. The condition when yields . Thus, (11) takes the formEquation (12) is expressed as Simplifying the above expression we obtainedwhere Integration of (14) leads us towhere The solutions of the field equation (8) are expressed as follows: the energy density is obtained aswhere .

The pressure is given asThe bulk viscous stress is expressed aswhere and .

The cosmological constant is expressed aswhere and .

Now, let us start with our proposed form of deceleration parameter . The different form of deceleration parameter is evolved as a result of considered value of and , which is expressed in Table 1. We know that in present scenario our Universe is accelerating. Thus serial numbers (), (), (), and () of Table 1 exhibit accelerating model. Now we will discuss the deceleration parameter in serial numbers (), (), (), and () of Table 1. For the choice of , the deceleration parameter in serial number () and () of Table 1 reduces to and , respectively, which is discussed by Mishra and Chand [29]. They called this deceleration parameter as bilinear variable deceleration parameter. We will discuss the case where of serial number () of Table 1 and also serial numbers () and () of Table 1. According to the serial numbers (), (), and () of Table 1 we have three different models, which are discussed below.

3.1. Model I

The deceleration parameter in (6) for and takes the formHere we noticed that for and for , which means that our Universe is decelerating and accelerating in the provided ranges, respectively. Thus our Universe undergoes a phase transition from decelerating to accelerating phase.

For model I, the physical parameters are obtained as follows.

The Hubble parameter in (12) takes the formThe scale factor in (16) is expressed aswhere And The FRW space-time metric in (3) takes the form with the above mentation . The energy density , pressure , bulk viscous stress , and cosmological constant in (17), (18), (19), and (20) are expressed aswhere .where and .where and .