\left( \partial f / \partial R \right)^2}  = \frac{\Mp^2 M^2}{2} \frac{y \bar{f}_{,y} - \bar{f}}{ \bar{f}_{,y}^2} \, . \label{V} \end{aligned}$$ The potential of the inflaton field, $\phi$ is obtained from the specific form of $f(R)$ from the theory. Thus, one can analyze the usual inflationary behavior of this model in the EF.

One can obtain the slow-roll parameters as a function of $\bar{f}$ by using the Eq.(\[V\]) $$\begin{aligned}
 \epsilon_{\text{V}} &\equiv \frac{\Mp^2}{2} \left( \frac{V_{,\phi}}{V} \right)^{2} = \frac{\left(-2 \bff+y \bff_{,y} \right)^2}{3 \left(\bff - y \bff_{,y}\right)^2}\label{epsilon} \, ,  \\
\eta_{\text{V}} &\equiv \Mp^2 \frac{V_{,\phi\,\phi}}{V} = -\frac{2 \left(\bff_{,y}^2-4 \bff \bff_{,yy} +y \bff_{,y} \bff_{,yy} \right)}{3 \left( \bff -y \bff_{,y} \right) \bff_{,yy}}\label{eta} \, , \\
\xi_{\text{V}}^{2} &\equiv 2 \Mp^4 \frac{V_{,\phi} V_{,\phi\phi\phi}}{V^2} = \frac{8 \left(2 \bff -y \bff_{,y}\right) \left(3\bff_{,y}^2 \bff_{,yy}^2-8 \bff \bff_{,yy}^3+y \bff_{,y} \bff_{,yy}^3+\bff_{,y}^3 \bff_{,yyy}\right)}{9 \left(\bff-y \bff_{,y}\right)^2 \bff_{,yy}^3}\label{xi} \, , \end{aligned}$$ where $V_{,\phi}$ means the derivative of $V$ with respect to the inflaton field, $\phi$. The second expressions in Eqs.(\[epsilon\])-(\[xi\]) are described by the dimensionless quantities and one can use these formulae without any mistake in units. The scalar spectral index $n_{s}$, the tensor-to-scalar ratio $r$, the running of the spectral index $d n_{s} /d \ln k$ and the scalar primordial amplitude $A_{s}$ are obtained from the above slow-roll 