\nu}^{(R)} + T_{\mu \nu}^{(m)}\right) ~~,$$ where $$8 \pi G~ T_{\mu \nu}^{(R)} = \frac{1}{2} g_{\mu \nu} f(R) - R_{\mu \nu} f'(R) + \left(\nabla_{\mu}\nabla_{\nu} - g_{\mu \nu} \Box\right) f'(R)~~.$$ Here $f(R) = \frac{\partial f(R)}{\partial R}$, $R_{\mu \nu}$ is the Ricci tensor and $T_{\mu \nu}^{(m)}$ is the energy momentum tensor of matter and $R$ denotes the curvature contribution.

For a spatially flat FRW universe the modified Friedmann equation can be written as $$H^2 = \frac{8\pi G}{3} (\rho_m + \rho_R)$$ and $$2\dot{H} + 3H^2 = - 8\pi G (p_m +p_R)$$ where $$\label{maineq}
\rho_R = \frac{1}{8\pi G} \left[-\frac{1}{2}f(R) + 3(\dot{H} + H^2)f'(R) - 18(4H^2\dot{H} + H\ddot{H}) f''(R) \right] ~~,$$ $$\begin{aligned}
p_R = \frac{1}{8\pi G} &\bigg[\frac{1}{2}f(R) - (\dot{H} + 3H^2)f'(R) + 6(8H^2\dot{H} + 6H\ddot{H} + 4\dot{H}^2 
+ \stackrel{...}{H}) f''(R) \nonumber \\
& + 36(\ddot{H} + 4H\dot{H})^2 f'''(R) \bigg] ~~\end{aligned}$$ and $$\label{eqR}
R = 6(\dot{H} + 2H^2)~~.$$ Here the Hubble parameter is $H=\frac{\dot{a}}{a}$ and overdot denotes derivative with respect to cosmic time $t$. We can show that the curvature contribution will have its own equation of state and it can be written as [@f94] $$\begin{aligned}
\label{eos}
\omega_R &= \frac{p_R}{\rho_R} \nonumber \\
&= 1 - \frac{4\big[\dot{H}f'(R) + 3(3H\ddot{H} - 4H^2\dot{H} + 4\dot{H}^2 + \stackrel{...}{H})f''(R) + 18(\ddot{H} + 4H\dot{H})^2 