Bulk viscous cosmological model in Brans Dicke theory with new form of time varying deceleration parameter

In this article 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. Further, we have discussed the physical behaviors of the models.

the matter [1]. In the BDT, gravitational constant G is treated as the reciprocal of a massless scalar field φ, where φ is expected to satisfy a scalar wave equations and it's 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 R(t) 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] generalize 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 about the limit of the Brans-Dicke theory of gravity when ω → ∞ and shown 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 (ω ≥ 500) [6,7]. A more recent bound on the Brans-Dicke parameter ω is ω > 3300 [7]. A number of researchers [8,9,10,11,12,13,14,15] 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 the acceleration and contribute an energy density of same order of magnitude than 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. Singh & Singh [19] presented a cosmological model in BDT by considering cosmological constant as a function of scalar field φ. Exact cosmological solutions in BDT with uniform cosmological constant has been studied by Pimentel [20]. A class of flat FRW cosmological models with cosmological constant in BDT have also been obtained by Ahmadi & Riazi [21]. The age of the Universe from a view point of the nucleosynthesis with Λ term in BDT was investigated by Etoh et al. [22]. Azad & Islam.
[23] extended the idea of Singh & Singh [19] to study cosmological constant in Bianchi type I modified Brans-Dicke cosmology. Qiang [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 explore many dynamics of the homogeneous cosmological models. The bulk viscosity coefficient determine 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 [28] and references there in).
Motivated by the above studies, here we have discussed the variable cosmological constant Λ for FRW metric in the context of BDT with a special form of deceleration parameter.

Field equations
The field equation of Brans-Dicke theory in presence of cosmological constant may be written as where φ is the scalar field. The energy-momentum tensor T ij of the cosmic fluid in the presence of bulk viscosity may be be defined as Let us consider a homogeneous and isotropic Universe represented by FRW spacetime metric as where k (= 1, 0, −1) is the curvature parameter, which represents closed, flat and open model of the Universe and R(t) is the scale factor.
The FRW metric (4) and energy-momentum tensor (3) 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 (5)- (7), one must ensure that set of equations should be closed. Thus, two more physically reasonable relations are required amongst the variables.
First we consider a well accepted power law relation between scale factor R(t) and scalar field φ of the form [27] φ = φ 0 R α 1 (8) and as it has been well established that 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 form as the second physically plausible relation. Where α 2 , α 3 ∈ R. The considered form of deceleration parameter is motivated by the bilinear form of deceleration parameter [32]. 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 change andṘ is constant, the scaling factor is proportional to time, which leads to zero deceleration. In case when H is constant, the deceleration parameter (q) is also constant (-1). 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 [33]. We consider third physically plausible relation as the modified Chaplygin gas equation of state as follows [30,31] where A > 0, B > 0 are constants and 0 ≤ n ≤ 1.
The set of field equations (5)-(7) with the help of (8) may be written as (2 + α 1 )R R + 2 + 2α 1 + 2α 2 1 + ωα 2 Equations (11), (12) and (13), leads us to This equation is useful for obtaining the various cosmological solutions. Now our problem is to evaluate the R(t), which is obtained from the relation With the help of equation (9) and integrating (15), we obtained where c 1 is a constant of integration. The condition H → ∞ when t → 0 yields c 1 = 0. Thus, equation (16) takes the form Equation (17) is expressed as Simplifying the above expression we obtained where Integration of (18) leads us to where . The solutions of the field equation (11)-(13) is expressed as follows: The energy density ρ is obtained as where The bulk viscous stress Π is expressed as where where

S. No. Possible value of α 2 and α 3
Form of deceleration parameter q Behaviour of Cosmological model Phase trasition from decelerating to accelerating Phase trasition from accelerating to decelerating Accelerating of α 2 = α = α 4 , the deceleration parameter q in serial number 2 and 5 of Table 1 reduces respectively, which is discussed by [32]. They called this deceleration parameter as Bilinear variable deceleration parameter. We will discuss the case where α 2 = α = α 4 of serial number 5 of Table 1 and also serial number 8 and 9 of Table.1.
According to the serial number 5, 8 and 9 of Table 1 we have three different models, which are discussed below.

Model-I
The deceleration parameter q in (9) for α 2 > 0 and α 3 < 0 takes the form Here we noticed that, q > 0 for 0 < t < α 2 α 4 and q < 0 for t > α 2 α 4 , 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 (17) takes the form The scale factor R(t) in (19) is expressed as where . and The FRW space-time metric in (4) takes the form 1+α 2 e 2T 1 (t) dr 2 1 − kr 2 + r 2 dθ 2 + sin 2 θdφ 2 with the above mentation k i ,(i = 0, 1, 2, 3, 4). The energy density (ρ), pressure (p), bulk viscous stress (Π) and cosmological constant (Λ)in (20), (21), (22) and (23) are expressed as where where where Λ 1 = −0.5wα 2 1 + w(α 2 − 2)α 1 − 3(α 2 − 1) and Λ 2 = −0.5wα 2 1 − w(α 4 + 2)α 1 + 3(α 4 + 1). Figure 1 and Figure 2 represents the variation of deceleration parameter against time with    Here we observed that −1 < q < 1 for 0 < α 2 ≤ 1 and 0 < α 4 ≤ 1, which means that with in the provided range of α i (i = 2, 4) our Universe undergoes an accelerating power expansion. It can be observed from Figure 1 and     Figure 5 and Figure 6 represents the variation of energy density ρ and pressure p against time respectively for model-I. From the Figure 5 we pointed out that, in the interval 0 < α 4 ≤ 0.8 & α 2 = 0.5 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 α 4 ≥ 0.9 and α 2 = 0.5 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 interval 0 < α 4 ≤ 0.8 & α 2 = 0.5, the pressure is negative for small interval of time and increases with the evolution of time. In the interval 0.9 ≤ α 4 ≤ 1.2 & α 2 = 0.5, pressure is negative, which follow the observational data but for α 4 > 1.2 , it is complex valued, thus we neglect it.

Model-II
The deceleration parameter q in (9) for α 2 < 0 and α 3 < 0 takes the form Here we noticed that, q < 0 for α 4 , α 5 > 0, 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 (17) takes the form The scale factor R(t) in (19) is expressed as where T 1 (t) = k 0 t + k 1 . and The FRW space-time metric in (4) takes the form with the above mentation k i , (i = 0, 1, 2, 3, 4). The energy density (ρ), pressure (p), bulk viscous stress (Π) and cosmological constant (Λ)in (20), (21), (22) and (23) takes the form where where where Λ 1 = −0.5wα 2 1 − w(α 5 + 2)α 1 + 3(α 5 + 1) and Λ 2 = −0.5wα 2 1 − w(α 4 + 2)α 1 + 3(α 4 + 1). Now we will discuss about the physical parameters of the model-II. Figure 9 and Figure 10 represents the variation of deceleration parameter against time for fixed α 5 & different α 4 and fixed α 4 & different α 5 respectively. Here we observed that, deceleration parameter is negative and our model is accelerating.  • Hubble parameter H is a decreasing function of time and tending to zero with the evolution of time. As a representative case, we have presented for α 5 = 0.5 and different α 4 (0 < α 4 ≤ 1.2) as in Figure 11.   it increases gradually with the evolution of time (see Figure 12). Similar qualitative behaviour is noticed for α 5 = 0.5 and different α 4 (see Figure 13).  Figure 14 and Figure 15. The observations are as follows: • Energy density gradually decreases and approaches towards zero with the evolution of • Energy density is gradually decreased for small interval of time and tends towards infinity with the evolution of time for 0.4 ≤ α 4 ≤ 0.9 and α 5 = 0.3.
• For α 4 ≥ 1 and α 5 = 0.3, energy density tends towards zero with time. Here we pointed out that, with the increment of α 4 the bounce of the energy density increases and gradually tending to zero (see Figure 14).
• Pressure is negative for a small interval of time & gradually increases with time and it takes values from positive to negative in the interval 0.4 ≤ α 4 ≤ 0.9 and α 4 ≥ 1.6 with α 5 = 0.3 respectively (see Figure 15).
The variation of bulk viscous stress Π and cosmological constant Λ against time for model-II is presented in Figure 16 and Figure 17 respectively. The observations are as follows: Bulk viscous stress Π(see Figure 16)

Model-III
The deceleration parameter q in (9) for α 2 < 0 and α 3 = 0 takes the form Here we noticed that, q < 0 for α 5 > 0, 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 (17) takes the form The scale factor R(t) in (19) is expressed as where T 1 (t) = k 0 t + k 1 . and The FRW space-time metric in (4) takes the form with the above mentation k i , (i = 0, 1, 2, 3, 4). The energy density (ρ), pressure (p), bulk viscous stress (Π) and cosmological constant (Λ)in (20), (21), (22) and (23) takes the form where where where Λ 1 = −[0.5wα 2 1 + w(α 5 + 2)α 1 − 3(α 5 + 1)] and Λ 2 = −0.5wα 2 1 − 2wα 1 + 3. The profile of deceleration parameter, Hubble parameter and scale factor against time is plotted in the Figure 18, Figure 19 and Figure    • Deceleration parameter q is 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 follow an expansion with constant rate at late time (see Figure 18).
• Hubble parameter H is a decreasing function of time and H → 0 when t → ∞. Also in this case higher the value of α 5 , higher is the value of Hubble parameter (see Figure 19).
• Scale factor R is an increasing function of time and R → ∞ when t → ∞. Equation (40) indicates that, R is not defined for α 5 = 1. As a representative case, we considered 0 < α 5 < 1 (see Figure 20). For α 5 ≥ 1, energy density possess physical unrealistic behavior, so α 5 is restricted to 0 < α 5 < 1. It is noticed that, energy density is a decreasing function of time and ρ → 0 when t → ∞ (see Figure 21). Also pressure is a negative quantity with the evolution of time (see Figure 22).

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 equation (9). We have presented three different cosmological models based on the choice of α 2 and α 3 . The physical parameters involved in these three models are physically acceptable for some interval of α 2 and α 3 , which follow the observational data. Here we would like conclude that, for physically acceptable cosmological models the choice of α 2 and α 3 are crucial.