matter to constrain its parameters.

General formalism
-----------------

The action for the general $f(R)$-gravity theories in 4 dimensional space-time can be rewritten as $$S \equiv \frac{1}{16 \pi G} \int \sqrt{-g} \, d^4 x f(R) \equiv \frac{\Mp^2}{2} \int \sqrt{-g} \, d^4 x M^2 \bar{f}(y) \, , \label{Sfx}$$ where $\Mp$ is the reduced Planck mass, $M$ is the characteristic mass scale, $y = R/M^2$ is the normalized dimensionless variable, and $\bar{f}$ is a dimensionless function. The above Jordan frame (JF) action is conformally transformed into the Einstein frame (EF) action by doing the Weyl transformation of the metric and the field redefinition $$\begin{aligned}
 S_{\text{E}} &= \int  \sqrt{-g^{\text{E}} } \, d^4 x \, \left( \frac{\Mp^2 }{2} R_{\text{E}} - \frac{1}{2} \partial^{\mu} \phi \partial_{\mu} \phi - V \left( \phi \right) \right) \label{ESR2lnR} \, , \\
& \text{where}  \,\,\,\,\,\, g^{(\text{E})}_{\mu \nu} \equiv \frac{\partial f}{\partial R} g_{\mu\nu} \, , \label{gE} \\ 
\phi  &\equiv \Mp \bphi = \sqrt{\frac{3}{2}} \Mp \ln \left[ \, \frac{\partial f }{\partial R} \, \right]  = \sqrt{\frac{3}{2}} \Mp \ln \left[ \, \frac{\partial \bar{f} }{\partial y} \, \right] \equiv \sqrt{\frac{3}{2}} \Mp \ln \bar{f}_{,y} \left( y \right) \, , \label{bphi} \\ 
V \left( \phi \right) &\equiv \frac{\Mp^2}{2} \frac{R \left( \partial f / \partial R \right) - f}{ 