#### 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.

The matter energy density parameter and dark energy density parameter are given byUsing (40) and (42), we obtain overall density parameter asThe variation of density parameters and with redshift is depicted in Figure 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 (Figure 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.

##### 3.2. Interacting Dark Energy and Dark Matter

In this section, we have considered interaction between DE and DM. For this purpose we can write the energy conservation equation (30) as andwhere is interacting term.

To find the solution of coincidence problem, we have considered an energy transfer from dark energy to dark matter by assuming , which ensures that the second law of thermodynamics is fulfilled (Pavon and Wang [55]). The continuity equations (45) and (46) imply that the interaction term should be proportional to inverse of time. Therefore, a first and natural candidate can be the Hubble parameter multiplied with the energy density. Following Amendola et al. [56] and Guo et al. [57], we considerwhere is coupling coefficient which can be considered as a constant or function of redshift .

Using (45), we obtain the energy density of dark matter aswhere is an integrating constant.

Using (48) in (29), we obtain the energy density of DE asUsing (48) and (50) in (10), we obtainThis is the general form of the EoS parameter of DE in BD theory for interacting case. Here is the deceleration parameter.

Now using (16) in (52), we obtain the EoS parameter in terms of redshift asThe behavior of EoS parameter in terms of redshift is depicted in Figure 3. We fixed the parameters , and vary as 0.01, 0.05, and 0.09, respectively. This figure shows that, for all values of coupling constant , the EoS parameter of DE is varying in quintessence region, crosses the PDL, and varies in phantom region. At late time (i.e., at ), the EoS parameter of DE . Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.

The expression for matter energy density and dark energy density are given byandUsing (55) and (57), we obtain total energy density parameterand this equation is same as (44). The variation of density parameter and with redshift is depicted in Figure 4. In this 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 is, the more effectively structures will have been formed in the past.

#### 4. Physical Acceptability and Stability Analysis

To find the stability condition of corresponding models, we use squared speed of sound (). A positive value of squared speed of sound () represents a stable model whereas the negative value of squared speed of sound () indicates the instability of model. A squared speed of sound () is defined asThe squared speed of sound for non-interacting and interacting models are, respectively, given byand From Figures 5 and 6, it is observed that in our non-interacting and interacting models the sound speed remains positive (i.e., ); hence our models shows the stability throughout the evolution of the universe.

Secondly, the plot of weak energy condition (WEC), dominant energy condition (DEC), and strong energy condition (SEC) for non-interacting and interacting cases is shown in Figures 7 and 8, respectively. It is observed that, in non-interacting and interacting cases, the energy conditions obey the following restrictions:(i),(ii),(iii).

From Figures 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.

#### 5. Anisotropy Parameter

The anisotropy parameter of expansion is defined asThe anisotropy parameter versus redshift is depicted in Figure 9. It is observed that the anisotropy parameter is decreasing function and dies out at late time . Hence the model reaches isotropy which matches with the recent observations as the universe is isotropic at large scale.

#### 6. Statefinder Parameters

In order to get an accurate analysis to discriminate among the dark energy models Sahni et al. [58] proposed a new geometrical diagnostic named as statefinder pair which is constructed from scale factor () and its derivative up to third order. The statefinder parameters are defined asThese parameters allow us to characterize the properties of DE. Using these parameters one can describe the well-known region as follows: indicates limit and indicates CDM limit, while and correspond to region of phantom and quintessence dark energy era. The relation between statefinder parameters for our models is obtain asFrom (64) and (65) a relation between parameters* r* and* s* is given byThe variation of parameter versus is plotted in Figure 10.

From Figure 9, it is seen that the curve passes through the point ; thus it can be concluded that our model corresponds to model.

#### 7. 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 ‘’ 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 physically acceptable and stable.

In non-interacting case, the EoS parameter of DE is varying in quintessence region and crossing PDL depending on the values of constant whereas in interacting case for all small values of coupling constant the EoS parameter of DE varies in quintessence region; it crosses the PDL and varies in phantom region. However, at late time (i.e., at ) the EoS parameter of both the cases tends to a cosmological constant . The anisotropy parameter of expansion is calculated and studied. It is observed that for (i.e., at initial time) it is infinite but at (i.e., at present time) it is closer to zero and at it completely dies out which matches with the present day observations, which indicate that our model at present time () is closer to the de Sitter model and attains complete de Setter model at ; 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 is adapted to differentiate the different forms of DE. The trajectories in the plane corresponds to the model (as shown in Figure 10).

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.