On inflationary parameters in scalar-tensor theories

The equivalence between $f(R)$ and scalar-tensor theories is revisited, we consequently explored different $f(R)$ models. After consideration of specific definition of the scalar field, we derived the potentials $V(\phi)$ for each $f(R)$ model focusing on the early Universe, mostly the inflation epoch. For a given potential, we applied the slow-roll approximation approach to each $f(R)$ model and obtained the expressions for the spectral index $n_{s}$ and tensor-to-scalar ratio $r$. We determined the corresponding numerical values associated with each of the $f(R)$ models. Our results showed that for certain choice of parameter space, the values of $n_{s}$ and $r$ are consistent with the Planck survey results and others produce numerical values that are in the same range as suggested by Planck data. We further constructed the Klein-Gordon equations $(KGE)$ of each $f(R)$ model. We found numerical solutions to each KGE considering different values of free parameters and initial conditions of each $f(R)$ model. All models showed that the scalar field decreases as time increases, an indication that there is less content of the scalar field in the late Universe.