Anisotropic holographic dark energy model in Bianchi type-VI₀ universe in a scalar–tensor theory of gravitation

Abstract

In this paper, we investigate Bianchi type VI₀ universe filled with two minimally interacting fields, matter and anisotropic holographic dark energy components in the scalar–tensor theory of gravitation proposed by Saez and Ballester (Phys. Lett. A 113: 467, 1986). Solving the field equations of the theory using a relation between metric potentials and special law of variation for Hubble’s parameter proposed by Bermann (Nuovo Cimento B 74:182, 1983) we have presented an anisotropic holographic dark energy model in this theory. The physical aspects of the model are also discussed.

Appendix

Equation (4) is $(T^{ij} + \bar{T}^{ij})_{;j} = 0$, where

$$\begin{aligned} T^{ij}_{;j} =& \frac{1}{\sqrt{ - g}} \frac{\partial}{ \partial x^{j}} \bigl[ \sqrt{ - g} T^{ij} \bigr] + T^{\alpha \beta} \left \{ \textstyle\begin{array} {ccc}& i& \\ \alpha &&\beta \end{array}\displaystyle \right \} \\ =& \frac{1}{\sqrt{ - g}} \frac{\partial}{\partial x^{0}} \bigl[ \sqrt{ - g} T^{00} \bigr] + T^{00}\left \{ \textstyle\begin{array} {ccc} &0& \\ 0 &&0 \end{array}\displaystyle \right \} \\ &{} + T^{11}\left \{ \textstyle\begin{array} {ccc}& 0& \\ 1&& 1 \end{array}\displaystyle \right \} + T^{22}\left \{ \textstyle\begin{array} {ccc}& 0 & \\ 2 &&2 \end{array}\displaystyle \right \} + T^{33}\left \{ \textstyle\begin{array} {ccc} &0& \\ 3&& 3 \end{array}\displaystyle \right \} \end{aligned}$$

Here

$$\begin{aligned} &{ T_{0}^{0} = - \rho_{ M}, \qquad T_{1}^{1} = T_{2}^{2} = T_{3}^{3} = 0 }\\ &{ \bar{T}_{0}^{0} = - \rho_{\varLambda},\qquad \bar{T}_{1}^{1} = \omega \rho_{\varLambda} }\\ &{ \bar{T}_{2}^{2} = (\omega + \delta )\rho_{\varLambda},\qquad \bar{T}_{3}^{3} = (\omega + \gamma )\rho_{\varLambda}} \\ &{\bar{T}_{i}^{j} = \mathrm{diag}[ - 1, \omega, \omega + \delta, \omega + \gamma ]\rho_{\varLambda}} \end{aligned}$$

Similarly $\bar{T}^{ij}_{;j}$ can be defined. Now using Eqs. (5) and (6), we get Eq. (13). Since we are considering minimally interacting fields we have $T^{ij}_{;j} = 0$, $\bar{T}^{ij}_{;j} = 0$ so that, we get Eqs. (14) and (15) for metric (1).