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Advances in High Energy Physics
Volume 2017, Article ID 1390572, 24 pages
https://doi.org/10.1155/2017/1390572
Research Article

Bulk Viscous Cosmological Model in Brans-Dicke Theory with New Form of Time Varying Deceleration Parameter

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India

Correspondence should be addressed to Binaya K. Bishi; moc.liamg@cbyanib

Received 25 March 2017; Accepted 22 May 2017; Published 30 July 2017

Academic Editor: Shi-Hai Dong

Copyright © 2017 G. P. Singh and Binaya K. Bishi. 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 SCOAP3.

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 [815] 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.

Table 1: Different forms of deceleration parameter depending on the parameters and and behaviour of cosmological models according to the deceleration parameter .
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 .

Figures 1 and 2 represent the variation of deceleration parameter against time with different values of parameters as presented in the figures for model I. From these figures, we have noticed that when is fixed and is different and vice versa, deceleration parameter is a decreasing function of time and it takes values from positive to negative, which shows that our Universe undergoes a phase transition from deceleration phase to acceleration phase. Here we observed that for and , which means that within the provided range of our Universe undergoes an accelerating power expansion. It can be observed from Figures 1 and 2.

Figure 1: Variation of deceleration parameter against time for fixed and different
Figure 2: Variation of deceleration parameter against time for fixed and different

The variation of Hubble parameter and scale factor against time is plotted in Figures 3 and 4, respectively for model I. As a representative case here we have presented the variation of and for fixed and different as in figures. It is found that Hubble parameter is a decreasing function of time and approaching towards zero with the evolution of time. For and , the scale factor is an increasing function of time and the higher the value of , the lower the value of scale factor . For and , the scale factor takes a bounce and increases with the evolution of time (see Figure 4).

Figure 3: Variation of Hubble parameter against time for fixed and different
Figure 4: Variation of scale factor against time for fixed and different

Figures 5 and 6 represent the variation of energy density and pressure against time, respectively for model I. From the Figure 5 we pointed out that, in the interval & with the time, energy density decreases for small interval of time and increases to a higher value with the evolution of time. This shows that our Universe is dominated by radiation. For and the energy density is a decreasing function of time and approaches to zero with the evolution of time. In present scenario such type of qualitative behaviour of energy density is observed from observational data. From pressure profile (Figure 6) we observed that, in the intervals and , the pressure is negative for small interval of time and increases with the evolution of time. In the intervals and , pressure is negative, which follows the observational data but, for , it is complex valued; thus we neglect it.

Figure 5: Variation of energy density against time for , , , , , and different
Figure 6: Variation of pressure against time for , , , , , , , and different

The variation of bulk viscous stress and cosmological constant against time is plotted in Figures 7 and 8, respectively for model I. Figures indicate the qualitative and quantitative behaviour of both the parameters for open , flat , and closed Universe. We have noticed the following points.

Figure 7: Variation of bulk viscous stress against time for , , , , , , , and different . (b) shows the zooming of (a).
Figure 8: Variation of cosmological constant against time for , , , , and different . Lower two rows represent the magnified portion of the above two rows.

Bulk Viscous Stress  Π  (see Figure 7)(i)Bulk viscous stress takes values from positive to negative and approaches to minus infinity with time in case of flat and closed Universe whereas it is negative-positive-negative valued for open Universe in the intervals and .(ii)Bulk viscous stress is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative-positive valued for open Universe in and .(iii)For and , bulk viscous stress is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative valued for open Universe.Cosmological Constant  Λ  (see Figure 8)(i)Cosmological constant is positive and negative for flat and open Universe and closed Universe, respectively. Cosmological constant when .(ii)In case of flat and open Universe cosmological constant is positive valued for and whereas it is negative valued for open Universe.(iii)In case of flat Universe cosmological constant when but for close and open Universe when and when , respectively.

3.2. Model II

The deceleration parameter in (6) for and takes the formHere we noticed that for , which means that our Universe is accelerating with the evolution of time.

For model II, 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) takes the formwhere .where and .where and .

Now we will discuss the physical parameters of the model II. Figures 9 and 10 represent the variation of deceleration parameter against time for fixed and different and fixed and different , respectively. Here we observed that deceleration parameter is negative and our model is accelerating.

Figure 9: Variation of deceleration parameter against time for fixed and different
Figure 10: Variation of deceleration parameter against time for fixed and different

Figures 11, 12, and 13 depict the variation of Hubble parameter and scale factor against time, respectively, for model II. The observations are as follows:(i)Hubble parameter is a decreasing function of time and tending to zero with the evolution of time. As a representative case, we have considered and different as in Figure 11.(ii)Scale factor increases with the evolution of time. Here we pointed out that the qualitative behaviour of scale factor is different for different interval of and . As a representative case, we choose   and different and all other parameters as in Figures 12 and 13. In the interval & and , scale factor increases after taking a bounce where as in and , it increases gradually with the evolution of time (see Figure 12). Similar qualitative behaviour is noticed for and different (see Figure 13).

Figure 11: Variation of Hubble parameter against time for fixed and different
Figure 12: Variation of scale factor against time for fixed and different
Figure 13: Variation of scale factor against time for fixed and different

The variation of energy density and pressure against time is presented for model II in Figures 14 and 15, respectively. As a representative case we choose   and different and all other parameters are as in Figures 14 and 15. The observations are as follows:(i)Energy density gradually decreases and approaches towards zero with the evolution of time for and .(ii)Energy density is gradually decreased for small interval of time and tends towards infinity with the evolution of time for and .(iii)For and , energy density tends towards zero with time. Here we pointed out that, with the increment of , the bounce of the energy density increases and gradually tends to zero (see Figure 14).(iv)Pressure is negative in and with .(v)Pressure is negative for a small interval of time and gradually increases with time and it takes values from positive to negative in the intervals and with , respectively (see Figure 15).

Figure 14: Variation of energy density against time for , , , , , and different . (b) shows the zooming of (a).
Figure 15: Variation of pressure against time for , , , , , , , and different . (b) shows the zooming of (a).

The variation of bulk viscous stress and cosmological constant against time for model II is presented in Figures 16 and 17, respectively. The observations are as follows.

Figure 16: Variation of bulk viscous stress against time for , , , , , , , and different . (b) shows the zooming of (a). Blue line, Green line, and Red line represent , , and , respectively.
Figure 17: Variation of cosmological constant against time for , , , , and different . (b) shows the zooming of (a). Blue line, Green line, and Red line represent , , and , respectively.

Bulk Viscous Stress  Π  (see Figure 16)(i)It is positive valued for flat and closed Universe whereas it is negative valued for open Universe in and and .(ii)It is positive-negative valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in and .(iii)It is positive valued for flat and closed Universe whereas it is negative-positive valued for open Universe in and . Also it approaches towards infinity with the evolution of time in the specified interval of .(iv) is positive valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in and .Cosmological Constant  Λ  (see Figure 17)(i)For and , is positive valued for flat and closed Universe whereas it is negative value for open Universe. In case of flat Universe when but for open and closed Universe and with time respectively.(ii)For and , when for open, flat, and closed Universe. In case of flat and closed Universe, cosmological constant is positive valued whereas in open Universe it is negative-positive valued.(iii)It is positive valued for flat and closed Universe but it is negative-positive-negative valued for open Universe in and .

3.3. Model III

The deceleration parameter in (6) for and takes the formHere we noticed that for , which means that our Universe is accelerating with the evolution of time.

For model III, 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) takes the formwhere .where and .where and .

The profile of deceleration parameter, Hubble parameter, and scale factor against time is plotted in the Figures 18, 19, and 20, respectively, for model III. The observations are as follows:(i)Deceleration parameter is a negative valued function of time and approaches towards zero with the evolution of time. In other words we can say, at early time, our Universe is accelerating and follows an expansion with constant rate at late time (see Figure 18).(ii)Hubble parameter is a decreasing function of time and when . Also in this case, the higher the value of , the higher the value of Hubble parameter (see Figure 19).(iii)Scale factor is an increasing function of time and when . Equation (39) indicates that is not defined for . As a representative case, we considered (see Figure 20).Figures 21 and 22 depict the energy density and pressure profile against time. respectively. For , energy density possesses physical unrealistic behaviour, so is restricted to . It is noticed that energy density is a decreasing function of time and when (see Figure 21). Also pressure is a negative quantity with the evolution of time (see Figure 22).

Figure 18: Variation of deceleration parameter against time for different .
Figure 19: Variation of Hubble parameter against time different .
Figure 20: Variation of scale factor against time for different .
Figure 21: Variation of energy density against time for , , , , and different .
Figure 22: Variation of pressure against time for , , , , , , , and different . (b) shows the zooming of (a).

The profile of bulk viscous stress and cosmological constant against time is depicted in Figures 23 and 24, respectively, for model III. Bulk viscous stress is positive valued for flat and closed Universe whereas it is negative valued for open Universe. Similar quantitative behaviour is observed for cosmological constant. In case of flat Universe, cosmological constant is a decreasing function of time and tends to zero with the evolution of time.

Figure 23: Variation of bulk viscous stress against time for , , , , , , , and different . (b) shows the zooming of (a).
Figure 24: Variation of cosmological constant against time for , , , and different . (b) shows the zooming of (a).

4. Final Statements

In this article, we have studied the FRW cosmological model with modified Chaplygin gas in the framework of Brans-Dicke theory. The approximated exact solution is obtained for modified Einstein’s field equation with the help of proposed form of deceleration parameter as in (6). We have presented three different cosmological models based on the choice of and . The physical parameters involved in these three models are physically acceptable for some interval of and , which follow the observational data. Here we would like conclude that, for physically acceptable cosmological models, the choice of and is crucial.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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