Reexamining RHDE models in FRW Universe with two IR cutoff with redshift parametrization

Abstract

In this paper, we have investigated that the gravitational field equations are not compatible with conservation equation in Dixit et al. (Euro Phys J Plus 135:831, 2020). Therefore, the expression for an equation of state parameter along with the dynamics of $\omega _{T}$–$\omega _{T}^{\prime }$ plane does not reflect the actual behaviors of RHDE models in f(R, T) gravity and thus the method and technique given in Dixit et al. (2020) represents a fractured way for analyzing RHDE models in f(R, T) theory of gravity. We also investigate that the derived Universe is in decelerating phase of expansion for $0 < \beta \le 1.5$ which is contrary to the result obtained in Dixit et al. (2020).

Appendix

The Einstein field equation with RHDE fluid is read as

$$\begin{aligned} R_{ij} - \frac{1}{2}g_{ij}R = 8\pi \left( T_{ij}^{m} + T_{ij}^{\Re }\right) . \end{aligned}$$

(23)

where $T_{ij}^{m}$ and $T_{ij}^{\Re }$ are energy momentum tensor of matter and RHDE fluid, respectively.

Thus, the field equations for metric (6) are read as

$$\begin{aligned} 3H^{2}&= 8\pi (\rho + \rho _{T}), \end{aligned}$$

(24)

$$\begin{aligned} 2\dot{H} + 3H^{2}&= -8\pi (p + p_{T}). \end{aligned}$$

(25)

where $\rho$, p, $\rho _{T}$, and $p_{T}$ denote energy density of matter, the pressure of the matter, the energy density of RHDE fluid and pressure of RHDE fluid, respectively.

Differentiating Eq. (25) with respect to time and combining the resulting equation with Eq. (22), we obtain the following energy conservation equation

$$\begin{aligned} \dot{\rho } +3(\rho + p)H +\dot{\rho _{T}} + 3(\rho _{T} + p_{T})H = 0. \end{aligned}$$

(26)

Moreover, the field equations for metric (6) in the framework of $f(R, T) = R + 2f(T)$ theory of gravity are read as

$$\begin{aligned} 3H^{2}&= \left( 8\pi + 2f_{T}\right) \rho + 2pf_{T} + f(T), \end{aligned}$$

(27)

$$\begin{aligned} 2\dot{H} + 3H^{2}&= -8\pi p + f(T). \end{aligned}$$

(28)

From Eqs. (27) and (28), we can pick out a dark energy component due to f(T), described by

$$\begin{aligned} \rho _{T}&= 2f_{T}\rho + 2p f_{T} + f(T), \end{aligned}$$

(29)

$$\begin{aligned} p_{T}&= -f(T). \end{aligned}$$

(30)

It is worthwhile to note that one can reconstruct f(R, T) gravity from RHDE model by defining RHDE density in IR cutoff as $\rho _{T} = \frac{3c_{1}^{2}}{8\pi L^{2}(1 + \pi \delta L^{2})}$ with $c_{1}$ being a numerical constant. The parameters $\delta$ and L denote a non-additive parameter and length of horizon, respectively.

Differentiating Eq. (27) with respect to time and combining the resulting equation with Eq. (28), we obtain

$$\begin{aligned}&(8\pi + 2f_{T})\dot{\rho } + 3\left[ (8\pi + 2f_{T})\rho + 8\pi p\right] H + 2(\rho + p)\dot{f_{T}} \nonumber \\&\quad + (2\dot{p} + 6pH + 1)f_{T} = 0. \end{aligned}$$

(31)

Since $f(T) = \xi T$ which leads $f_{T} = \frac{\partial f(T)}{\partial T} = \xi$ and $\dot{f_{T}} = 0$, therefore, Eq. (31) leads to

$$\begin{aligned} (8\pi + 2\xi )\dot{\rho } + 3\left[ (8\pi + 2\xi )\rho + 8\pi p \right] H +(2\dot{p} + 6pH + 1)\xi = 0. \end{aligned}$$

(32)

From Eq. (32), it is clear that for $\xi = 0$, Eq. (32) converts into the conservation equation of general relativity case as we have discussed in Sect. 3.