# Bianchi Type-I Dark Energy Cosmology with Power-Law Relation in Brans-Dicke Theory of Gravitation

Authors: S. D. Katore; D. V. Kapse
Journal: Advances in High Energy Physics
 (2018)

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## 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 theΛCDM 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.


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## Body

## 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) parameterωde=pde/ρde which is not necessarily constant, where pde is the pressure and ρde is the energy density of DE. The DE candidate which can simply explain the cosmic acceleration is a vacuum energy ωde=-1, which is mathematically equivalent to the cosmological constant (Λ). The other conventional alternatives, which can be described by minimally coupled scalar fields, are quintessence -1<ωde<-1/3, phantom ωde<-1, 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 < ωde< −0.62 and −1.33 < ωde < −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 ϕ 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 Δ 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.


## 2. The Metric and BD Field Equations

We consider the homogeneous and anisotropic Bianchi type-I universe as(1)ds2=-dt2+A2dx2+B2dy2+C2dz2,where the scale factors A,B, and C are functions of time t only.BD field equations for the combined scalar and tensor fields with8πG=c=1 are given by (Brans-Dicke [37], Reddy et al. [43], and Rao et al. [49]) (2)Rij-12gijR-ω-ϕ-2ϕ,iϕ,j-12gijϕ,kϕ,k-ϕ-1ϕi;j-gijϕ;k,k=ϕ-1Tij,where R is the Ricci scalar, Rij is the Ricci tensor, ϕ is the Brans-Dicke scalar field, ω- is the dimensionless constant, and Tij is the energy momentum tensor. The scalar fields satisfy the following equation: (3)ϕ;k,k=Tji3+2ω-.The energy momentum tensor is given by (4)Tji=Tjmi+Tjdei,where Tjmi and Tjdei are energy momentum tensors of DM and DE, respectively. These are given by(5)Tjmi=diag-ρm,pm,pm,pm,(6)=diag-1,ωm,ωm,ωmρmand(7)Tjdei=diag-ρde,pde,pde,pde,(8)=diag-1,ωde,ωde,ωdeρde,where ρm and ρde are energy densities of DM and DE, respectively. Similarly, pm and pde are the pressure of DM and DE, respectively, while ωm=pm/ρm and ωde=pde/ρde 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 by(9)A˙B˙AB+B˙C˙BC+A˙C˙AC-ω-2ϕ˙2ϕ2+A˙A+B˙B+C˙Cϕ˙ϕ=ρm+ρdeϕ,(10)B¨B+C¨C+B˙C˙BC+ω-2ϕ˙2ϕ2+ϕ¨ϕ+B˙B+C˙Cϕ˙ϕ=1ϕ-ωmρm-ωdeρde,(11)A¨A+C¨C+A˙C˙AC+ω-2ϕ˙2ϕ2+ϕ¨ϕ+A˙A+C˙Cϕ˙ϕ=1ϕ-ωmρm-ωdeρde,(12)A¨A+B¨B+A˙B˙AB+ω-2ϕ˙2ϕ2+ϕ¨ϕ+A˙A+B˙Bϕ˙ϕ=1ϕ-ωmρm-ωdeρde,and the wave equation is(13)ϕ¨+A˙A+B˙B+C˙Cϕ˙=ρm1-3ωm+ρde1-3ωde3+2ω-,where an overhead dot denotes differentiation with respect to t.


## 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 (a) and scalar field (ϕ) (Pimental [51], Johri and Desikan [52]) as(14)ϕ=αaβ,where α and β>0 are constants.To examine the general results, we assume that the average scale factor is a hyperbolic function of time as Amirhashchi [53](15)at=sinh⁡t,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(16)at=11+z,z=1sinht-1,where z is the redshift parameter and a=ABC1/3 is the average scale factor.From (10)-(12), we obtain(17)A¨A-B¨B+C˙CA˙A-B˙B+A˙A-B˙Bϕ˙ϕ=0,(18)B¨B-C¨C+A˙AB˙B-C˙C+B˙B-C˙Cϕ˙ϕ=0,(19)A¨A-C¨C+B˙BA˙A-C˙C+A˙A-C˙Cϕ˙ϕ=0,Solving (17)-(19), we obtain(20)A˙A-B˙B=k1ABCϕ,(21)B˙B-C˙C=k2ABCϕ,(22)A˙A-C˙C=k3ABCϕ,where k1, k2, and k3 are constants of integration.Equations (20)-(22) further reduce to(23)AB=d1exp⁡k1∫dtABCϕ,(24)BC=d2exp⁡k2∫dtABCϕ,(25)AC=d3exp⁡k3∫dtABCϕ,where d1, d2, and d3are constants of integration.Using (23)-(25), we can write the metric functions A,B, and C explicitly as(26)A=a1aexp⁡b1∫dta3ϕ,(27)B=a2aexp⁡b2∫dta3ϕ,(28)C=a3aexp⁡b3∫dta3ϕ,where a1=d1d21/3,  a2=d1-1d31/3,  a3=d2d3-1/3,  b1=k1+k2/3,  b2=k3-k1/3, and  b3=-k2+k3/3, which satisfies the relations a1a2a3=1 and b1+b2+b3=0.Using (26)-(28) in (9), we obtain(29)H2=ρm+ρdeDϕ+KDa-6ϕ-2,where K=-b1b2+b2b3+b1b3,  D=3+β-β2ω-/2, and H=1/3A˙/A+B˙/B+C˙/C are the Hubble parameters. For b1=b2=b3=0, the model reduces to the flat FRW model in BD theory.The energy conservation equationT;jij=0 is Tm;jij+Tde;jij=0 and is given by(30)ρ˙m+3a˙a1+ωmρm+ρ˙de+3a˙a1+ωdeρde=0.


### 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])(31)ρ˙m+3a˙a1+ωmρm=0and(32)ρ˙de+3a˙a1+ωdeρde=0.Using (31), we obtain the energy density of DM as(33)ρm=ρ0ma-31+ωm(34)=ρ0m1+z31+ωm,where ρ0m>0 is a constant of integration.Using (33) and (29), we obtain the energy density of DE as(35)ρde=DH2αaβ-ρ0ma-31+ωm-Kα-1a-β+6(36)=1+1+z2Dα1+z-β-3Ω0m1+z31+ωm-Kα-11+zβ+6,where Ωm=ρm/3H2 is the energy density of DM and 0 denotes the present value of Ωm.Using (14), (33), and (35) in (10), we obtain the EoS parameter of DE as(37)ωde=H2β+2q-β21+ω-/2+β+1αaβ+Kα-1a-β+6-3ωmΩ0mH2a-31+ωmH2Dαaβ-3Ω0ma-31+ωm-Kα-1a-β+6.where q=-a¨/aH2=-1/1+1+z2 is the deceleration parameter.Using (16) in (37), we obtain the EoS parameter of DE in terms of redshift as(38)ωde=-β+21/1+1+z2+β21+ω-/2+β+1α1+z-β-Kα-11+zβ+6/1+1+z2+3ωmΩ0m1+z31+ωmDα1+z-β-Kα-11+zβ+6/1+1+z2-3Ω0m1+z31+ωm.The behavior of EoS parameter of DE in terms of redshiftz is depicted in Figure 1. Here the parameter ωm is taken to be zero and vary constant K as 0.01, 0.05, and 0.09, respectively. From figure it is clear that, for all small values of K, the EoS of DE is varying in quintessence region and crossing Phantom Divide line (PDL) ωde=-1. However, it is observed that at late time (i.e., at z=-1) the EoS parameter ωde≈-1. Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.Figure 1
The plot of EoS parameterωde versus redshift z for α=1,  β=0.01,  Ω0m=0.3,  ωm=0,  ω-=2, and vary K=0.01, 0.05, 0.09, resp.The matter energy density parameterΩm and dark energy density parameter Ωde are given by(39)Ωm=ρm3H2(40)=Ω0m1+z31+ωm,(41)Ωde=ρde3H2(42)=Dα31+z-β-Kα-11+zβ+631+1+z2-Ω0m1+z31+ωm.Using (40) and (42), we obtain overall density parameter Ω as(43)Ω=Ωm+Ωde(44)=Dα31+z-β-Kα-11+zβ+631+1+z2.The variation of density parameters Ωm and Ωde with redshift z 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 (Ω) approaches to 1. Therefore the model predicts a flat universe at late time. It is interesting to note that the value of β (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.Figure 2
The plot of total energy densities versus redshiftz for α=1,  β=0.01,Ω0m=0.3,  ωm=0,  ω-=2, and K=0.01.


### 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 (45)ρ˙m+3a˙a1+ωmρm=Qand(46)ρ˙de+3a˙a1+ωdeρde=-Q,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 assumingQ>0, 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 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(47)Q=3Hσρm,where σ>0 is coupling coefficient which can be considered as a constant or function of redshift z.Using (45), we obtain the energy density of dark matter as(48)ρm=ρ0ma-31+ωm-σ(49)=ρ0m1+z31+ωm-σ,where ρ0m is an integrating constant.Using (48) in (29), we obtain the energy density of DE as(50)ρde=DH2αaβ-ρ0ma-31+ωm-σ-Kα-1a-β+6(51)=1+1+z2Dα1+z-β-3Ω0m1+z31+ωm-σ-Kα-11+zβ+6.Using (48) and (50) in (10), we obtain(52)ωde=H2β+2q-β21+ω-/2+β+1αaβ+Kα-1a-β+6-3H2ωmΩ0ma-31+ωm-σH2Dαaβ-3Ω0ma-31+ωm-σ-Kα-1a-β+6.This is the general form of the EoS parameter of DE in BD theory for interacting case. Here q=-a¨/aH2 is the deceleration parameter.Now using (16) in (52), we obtain the EoS parameter in terms of redshift as(53)ωde=-β+21/1+1+z2+β21+ω-/2+β+1α1+z-β-Kα-11+zβ+6/1+1+z2+3ωmΩ0m1+z31+ωm-σDα1+z-β-Kα-11+zβ+6/1+1+z2-3Ω0m1+z31+ωm-σ.The behavior of EoS parameter in terms of redshift z is depicted in Figure 3. We fixed the parameters ωm=0,K=0.01, and vary σ as 0.01, 0.05, and 0.09, respectively. This figure shows that, for all values of coupling constant σ, the EoS parameter of DE is varying in quintessence region, crosses the PDL, and varies in phantom region. At late time (i.e., at z=-1), the EoS parameter of DE ωde≈-1. Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.Figure 3
The plot of EoS parameterωde versus redshift z for α=1,β=0.01,  Ω0m=0.3,ωm=0,ω-=2, and vary K=0.01, 0.05, 0.09, resp.The expression for matter energy densityΩm and dark energy density Ωde are given by(54)Ωm=ρm3H2(55)=Ω0m1+z31+ωm-σ,and(56)Ωde=ρde3H2(57)=Dα31+z-β-Kα-11+zβ+631+1+z2-Ω0m1+z31+ωm-σ.Using (55) and (57), we obtain total energy density parameter(58)Ω=Ωm+Ωde=Dα31+z-β-Kα-11+zβ+631+1+z2,and this equation is same as (44). The variation of density parameter Ωm and Ωde with redshift z 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.Figure 4
The plot of total energy densities versus redshiftz for α=1,β=0.01,  Ω0m=0.3,ωm=0ω-=2,  K=0.01, and σ=0.18.


## 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])(31)ρ˙m+3a˙a1+ωmρm=0and(32)ρ˙de+3a˙a1+ωdeρde=0.Using (31), we obtain the energy density of DM as(33)ρm=ρ0ma-31+ωm(34)=ρ0m1+z31+ωm,where ρ0m>0 is a constant of integration.Using (33) and (29), we obtain the energy density of DE as(35)ρde=DH2αaβ-ρ0ma-31+ωm-Kα-1a-β+6(36)=1+1+z2Dα1+z-β-3Ω0m1+z31+ωm-Kα-11+zβ+6,where Ωm=ρm/3H2 is the energy density of DM and 0 denotes the present value of Ωm.Using (14), (33), and (35) in (10), we obtain the EoS parameter of DE as(37)ωde=H2β+2q-β21+ω-/2+β+1αaβ+Kα-1a-β+6-3ωmΩ0mH2a-31+ωmH2Dαaβ-3Ω0ma-31+ωm-Kα-1a-β+6.where q=-a¨/aH2=-1/1+1+z2 is the deceleration parameter.Using (16) in (37), we obtain the EoS parameter of DE in terms of redshift as(38)ωde=-β+21/1+1+z2+β21+ω-/2+β+1α1+z-β-Kα-11+zβ+6/1+1+z2+3ωmΩ0m1+z31+ωmDα1+z-β-Kα-11+zβ+6/1+1+z2-3Ω0m1+z31+ωm.The behavior of EoS parameter of DE in terms of redshiftz is depicted in Figure 1. Here the parameter ωm is taken to be zero and vary constant K as 0.01, 0.05, and 0.09, respectively. From figure it is clear that, for all small values of K, the EoS of DE is varying in quintessence region and crossing Phantom Divide line (PDL) ωde=-1. However, it is observed that at late time (i.e., at z=-1) the EoS parameter ωde≈-1. Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.Figure 1
The plot of EoS parameterωde versus redshift z for α=1,  β=0.01,  Ω0m=0.3,  ωm=0,  ω-=2, and vary K=0.01, 0.05, 0.09, resp.The matter energy density parameterΩm and dark energy density parameter Ωde are given by(39)Ωm=ρm3H2(40)=Ω0m1+z31+ωm,(41)Ωde=ρde3H2(42)=Dα31+z-β-Kα-11+zβ+631+1+z2-Ω0m1+z31+ωm.Using (40) and (42), we obtain overall density parameter Ω as(43)Ω=Ωm+Ωde(44)=Dα31+z-β-Kα-11+zβ+631+1+z2.The variation of density parameters Ωm and Ωde with redshift z 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 (Ω) approaches to 1. Therefore the model predicts a flat universe at late time. It is interesting to note that the value of β (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.Figure 2
The plot of total energy densities versus redshiftz for α=1,  β=0.01,Ω0m=0.3,  ωm=0,  ω-=2, and K=0.01.


## 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 (45)ρ˙m+3a˙a1+ωmρm=Qand(46)ρ˙de+3a˙a1+ωdeρde=-Q,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 assumingQ>0, 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 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(47)Q=3Hσρm,where σ>0 is coupling coefficient which can be considered as a constant or function of redshift z.Using (45), we obtain the energy density of dark matter as(48)ρm=ρ0ma-31+ωm-σ(49)=ρ0m1+z31+ωm-σ,where ρ0m is an integrating constant.Using (48) in (29), we obtain the energy density of DE as(50)ρde=DH2αaβ-ρ0ma-31+ωm-σ-Kα-1a-β+6(51)=1+1+z2Dα1+z-β-3Ω0m1+z31+ωm-σ-Kα-11+zβ+6.Using (48) and (50) in (10), we obtain(52)ωde=H2β+2q-β21+ω-/2+β+1αaβ+Kα-1a-β+6-3H2ωmΩ0ma-31+ωm-σH2Dαaβ-3Ω0ma-31+ωm-σ-Kα-1a-β+6.This is the general form of the EoS parameter of DE in BD theory for interacting case. Here q=-a¨/aH2 is the deceleration parameter.Now using (16) in (52), we obtain the EoS parameter in terms of redshift as(53)ωde=-β+21/1+1+z2+β21+ω-/2+β+1α1+z-β-Kα-11+zβ+6/1+1+z2+3ωmΩ0m1+z31+ωm-σDα1+z-β-Kα-11+zβ+6/1+1+z2-3Ω0m1+z31+ωm-σ.The behavior of EoS parameter in terms of redshift z is depicted in Figure 3. We fixed the parameters ωm=0,K=0.01, and vary σ as 0.01, 0.05, and 0.09, respectively. This figure shows that, for all values of coupling constant σ, the EoS parameter of DE is varying in quintessence region, crosses the PDL, and varies in phantom region. At late time (i.e., at z=-1), the EoS parameter of DE ωde≈-1. Therefore we say that the cosmological constant is a suitable candidate to represent the behavior of DE in the derived model at late times.Figure 3
The plot of EoS parameterωde versus redshift z for α=1,β=0.01,  Ω0m=0.3,ωm=0,ω-=2, and vary K=0.01, 0.05, 0.09, resp.The expression for matter energy densityΩm and dark energy density Ωde are given by(54)Ωm=ρm3H2(55)=Ω0m1+z31+ωm-σ,and(56)Ωde=ρde3H2(57)=Dα31+z-β-Kα-11+zβ+631+1+z2-Ω0m1+z31+ωm-σ.Using (55) and (57), we obtain total energy density parameter(58)Ω=Ωm+Ωde=Dα31+z-β-Kα-11+zβ+631+1+z2,and this equation is same as (44). The variation of density parameter Ωm and Ωde with redshift z 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.Figure 4
The plot of total energy densities versus redshiftz for α=1,β=0.01,  Ω0m=0.3,ωm=0ω-=2,  K=0.01, and σ=0.18.


## 4. Physical Acceptability and Stability Analysis

To find the stability condition of corresponding models, we use squared speed of sound (vs2). A positive value of squared speed of sound (vs2) represents a stable model whereas the negative value of squared speed of sound (vs2) indicates the instability of model. A squared speed of sound (vs2) is defined as(59)vs2=p˙deρ˙de.The squared speed of sound for non-interacting and interacting models are, respectively, given by(60)vs2=αβz+1-1-β1+1+z2β2ω-2+1+β+21+1+z2+β+1-α2z+21+z-ββ2ω-2+1+β+21+1+z2+β+1+αβ+22z+21+z-β1+1+z2+Kα-1β+61+zβ+5-3ωmΩ0m1+z3ωm+23ωm1+1+z2+5z2+10z+8/-αDz+1-1-β21+z2-β1+1+z2-Kα-1β+61+zβ+5-3Ω0m1+z3ωm+23ωm1+1+z2+5z2+10z+8and (61)vs2=αβz+1-1-β1+1+z2β2ω-2+1+β+21+1+z2+β+1-α2z+21+z-ββ2ω-2+1+β+21+1+z2+β+1+αβ+22z+21+z-β1+1+z2+Kα-1β+61+zβ+5-3ωmΩ0m3ωm-3σ+31+1+z21+z3ωm-3σ+2+2z+2z+13ωm-3σ+3/-αDz+1-1-β21+z2-β1+1+z2-Kα-1β+61+zβ+5-3Ω0m3ωm-3σ+31+1+z21+z3ωm-3σ+2+2z+2z+13ωm-3σ+3From Figures 5 and 6, it is observed that in our non-interacting and interacting models the sound speed remains positive (i.e., vs2>0); hence our models shows the stability throughout the evolution of the universe.Figure 5
The plot of sound speedvs2 versus redshift z in non-interacting case for α=1,β=1,Ω0m=0.3,ω-=1ωm=0, and vary K=0.01, 0.05, 0.09, resp.Figure 6
The plot of sound speedvs2 versus redshift z in interacting case for α=1,β=1,Ω0m=0.3,ω-=1,ωm=0,K=0.01, and vary σ=0.01, 0.05, 0.09, resp.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 Figures7 and 8, respectively. It is observed that, in non-interacting and interacting cases, the energy conditions obey the following restrictions:(i)
ρde≥0,(ii)
ρde+pde≥0,(iii)
ρde+3pde≤0.Figure 7
The plot of the weakρde≥0, dominant ρde+pde≥0, and strong ρde+3pde≥0 energy conditions versus redshift z for non-interacting scenario.Figure 8
The plot of the weakρde≥0, dominant ρde+pde≥0, and strong ρde+3pde≥0 energy conditions versus redshift z for interacting scenario.From Figures7 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 ParameterΔ

The anisotropy parameter of expansionΔ is defined as(62)Δ=13H2∑i=13Hi-H2=b12+b22+b323H2ϕ2a6=b12+b22+b321+z2β+33α21+1+z2.The anisotropy parameter versus redshift z is depicted in Figure 9. It is observed that the anisotropy parameter is decreasing function and dies out at late time z=-1. Hence the model reaches isotropy which matches with the recent observations as the universe is isotropic at large scale.Figure 9
The plot of anisotropy parameterΔ versus redshift z.


## 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 r,s which is constructed from scale factor (a) and its derivative up to third order. The statefinder parameters are defined as(63)r=a⃛aH3,s=r-13q-1/2.These parameters allow us to characterize the properties of DE. Using these parameters one can describe the well-known region as follows: r,s=(1,0) indicates ΛCDM limit and r,s=1,1 indicates CDM limit, while s>0 and r<1 correspond to region of phantom and quintessence dark energy era. The relation between statefinder parameters for our models is obtain as(64)r=11+1+z2,(65)s=21+z231+1+z2.From (64) and (65) a relation between parametersr ands is given by(66)s=21-r3.The variation of parameter s versus r is plotted in Figure 10.Figure 10
The plot of statefinder parameterss versus r.From Figure9, it is seen that the curve passes through the point r=1,s=0; thus it can be concluded that our model corresponds to ΛCDM 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 ‘ϕ’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 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 constantK whereas in interacting case for all small values of coupling constant σ 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 z=-1) the EoS parameter of both the cases tends to a cosmological constant ωde=-1. The anisotropy parameter of expansion Δ is calculated and studied. It is observed that for z=1 (i.e., at initial time) it is infinite but at z=0 (i.e., at present time) it is closer to zero and at z=-1 it completely dies out which matches with the present day observations, which indicate that our model at present time (z=0) is closer to the de Sitter model and attains complete de Setter model at z=-1; 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 ΛCDM model (as shown in Figure 10).



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*Source: 2854567-2018-07-09.xml*