Future cosmological evolution in $f(R)$ gravity using two equations of state parameters

TL;DR

This paper explores the future behavior of $f(R)$ gravity models, focusing on oscillations around the phantom divide and the emergence of singularities due to stability condition violations.

Contribution

It introduces two types of energy density and pressure in $f(R)$ gravity and analyzes their impact on future oscillations and singularities.

Findings

Future oscillations around the phantom divide are confirmed in $f(R)$ gravity.

Singularities occur when the stability condition of $f(R)$ gravity is violated.

Two types of equations of state are used to describe the $f(R)$-fluid.

Abstract

We investigate the issues of future oscillations around the phantom divide for $f(R)$ gravity. For this purpose, we introduce two types of energy density and pressure arisen from the $f(R)$-higher order curvature terms. One has the conventional energy density and pressure even in the beginning of the Jordan frame, whose continuity equation provides the native equation of state $w_{\rm DE}$. On the other hand, the other has the different forms of energy density and pressure which do not obviously satisfy the continuity equation. This needs to introduce the effective equation of state $w_{\rm eff}$ to describe the $f(R)$-fluid, in addition to the native equation of state $\tilde{w}_{\rm DE}$. We confirm that future oscillations around the phantom divide occur in $f(R)$ gravities by introducing two types of equations of state. Finally, we point out that the singularity appears ar $x=x_c$ because the stability condition of $f(R)$ gravity violates.