Bianchi type-I dark energy cosmology with power-law relation in Brans-Dicke theory of gravitation

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 by using (i) 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. {r, s} is adopted to differentiate our dark energy models.

The study of DE is possible through its equation of state (EoS) parameter de de 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 and quintom (that can across 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 < de  < −0.62 and −1.33< de  < −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 DM lead 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 can't 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][30][31][32][33], Adhav et al. [34,35], Fayaz [36]  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. {r, s} is adopted to characterize different phases of the universe in section 6 and finally, Section 7 contains some concluding remarks.

The metric and BD field equations
We consider the homogeneous and anisotropic Bianchi type-I universe as where the scale factors A,B and C are functions of time t only.
BD field equations for the combined scalar and tensor fields with   where R is the Ricci scalar, ij R is the Ricci tensor,  is the Brans-Dicke scalar field,  is the dimensionless constant and ij T is the energy momentum tensor. The scalar fields satisfy the following equation The energy momentum tensor is given by where  i m j T and  i de j T are energy momentum tensors of dark matter and dark energy, respectively. These are given by In co-moving coordinate system, the BD field equations (2) and (3) for the metric (1), are given by and the wave equation is where an overhead dot denotes differentiation with respect to t.

Solutions of field equations
We have initially six variables and four linearly independent equations (7)-(10). 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 (a) and scalar field (  ) (Pimental [51], Johri and Desikan (12) where  and 0   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 (q). 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 by where z is the redshift parameter and is the average scale factor.
Using equations (20)- (22), we can write the metric functions A, B and C explicitly as which satisfies the relations 1 and 0 Using equations (23)- (25) in (7), we obtain where   is the Hubble parameter. For , 0 the model reduces to the flat FRW model in BD theory.
The energy conservation equation and is given by

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 (27), leads to (Harko et al. [50]) and   0 Using equation (28), we obtain the energy density of DM as is a constant of integration.
Using equations (30) and (26), we obtain the energy density of DE as is the energy density of DM and 0 denotes the present value of m  .
Using equations (12), (30) and (32) in equation (8), we obtain the EoS parameter of DE as Using equation (13) in equation (33), we obtain the EoS parameter of DE in terms of redshift  The behavior of EoS parameter of DE in terms of redshift z is depicted in Fig. 1 Using equations (35) and (36), we obtain overall density parameter  as The variation of density parameters m  and de  with redshift z is depicted in Fig. 2. Dot denotes the current value of these parameters. It is observed that for sufficiently large time, the overall density parameter (  ) approaches to 1. Therefore the model predicts a flat universe at late time. It is interesting to note that the value of  (i. e. BD theory) brings impact on the evolution of the densities (Fig. 2). The dark energy density parameter is increasing whereas the matter density parameter is decreasing. It is also clear that the value of dark energy density parameter is greater than the matter density parameter. Thus, the universe is dominated by dark energy throughout the evolution.

Interacting Dark energy and
where Q is interacting term.
To find the solution of coincidence problem, we have considered an energy transfer from dark energy to dark matter by assuming Q>0, which ensures that the second law of thermodynamics is fulfilled (Pavon and Wang [55]). The continuity equation (38) and (39) implies that the interaction term Q should be proportional to inverse of time. Therefore, a first and natural candidate can be the Hubble parameter H multiplied with the energy density.
Following Amendola et al. [56] and Guo et al. [57], we consider where 0   is coupling coefficient which can be considered as a constant or function of redshift z.
Using equation (38), we obtain the energy density of dark matter as Using equation (41) in equation (26), we obtain the energy density of DE as Using equations (41) and (42) in equation (8), we obtain is the deceleration parameter. Now using equation (13) in equation (43), we obtain the EoS parameter in terms of redshift   1  3  0  2   6  1   1  1  1  3   1  1  3 .
Using equations (45) and (46), we obtain total energy density parameter this equation is same as equation (37). The variation of density parameter m  and de  with redshift z is depicted in Fig. 4. In figure the dot denotes the current value of these parameters.
Hence we observed that in interacting case the density parameter has the same properties as in non-interacting case. From Figures 2 and 4, we observed that with interaction DE and DM follow one another. This means that in the recent history of the universe DE is being transformed into DM and the fluctuations do get more effective in the past. Note that the stronger the interaction, the more effectively structures will have been formed in the past.

Physical acceptability and stability analysis
To find the stability condition of corresponding models, we use squared speed of sound ( 2 s v ).
Secondly, the plot of weak energy condition (WEC), dominant energy condition (DEC) and strong energy condition (SEC) for non-interacting and interacting cases as shown in Fig. 7 and Fig. 8 respectively. It is observed that in non-interacting and interacting cases, the energy conditions obey the following restrictions: From Figs. 7 and 8 and above expressions, we observed that the WEC and DEC are satisfied for non-interacting and interacting cases whereas the SEC is violated in entire evolution of the universe in non-interacting and interacting scenario. Therefore, on the basis of above discussion and analysis, our corresponding models are physically acceptable.
The variation of parameter s versus r is plotted in Fig. 10.  From Fig. 9, it is seen that the curve passes through the point   0 , 1   s r , thus it can be concluded that our model corresponds to CDM  model.

Conclusion
In this paper, we have studied interacting and non-interacting DE and DM in the anisotropic Bianchi type-I universe in the framework of Brans-Dicke theory of gravitation. To obtain the exact solutions of Brans-Dicke field equations we have used (i) the power-law relation between ' 'and 'a' and (ii) the average scale factor in terms of redshift. In non-interacting and interacting cases the general form of EoS parameter is derived. Then using the scale factor in terms of redshift, results are examined. We have discussed the physical acceptability and stability of our models. It is found that our models are physical acceptable and stable.
In .i.e. we can say that the Bianchi type-I space-time reduces to flat FRW (isotropic) soon after the inflation. As mentioned in Carroll et al. [59], this ensures that there is no Big rip singularity; rather, the universe eventually settles into a de Sitter phase. Finally the statefinder diagnostic pair {r, s} is adapted to differentiate the different forms of DE. The trajectories in the {r, s} plane corresponds to the CDM  model (as shown in Figure 10).