Generalized scalar-tensor theory of gravity reconstruction from physical potentials of a scalar field

We describe how to reconstruct generalized scalar-tensor gravity (GSTG) theory, which admits exact solutions for physical type of the potentials. Our consideration deals with cosmological inflationary models based on GSTG with non-minimal coupling of a (non-canonical) scalar field to the Ricci scalar. The basis of proposed approach to the analysis of these models is a priori specified relation between the Hubble parameter $H$ and a function of non-minimal coupling $F=1+\delta F$ as $H\propto\sqrt{F}$. Deviations from Einstein gravity $\delta F$ induce a corresponding deviations of the potential $\delta V$ from a constant value and modify the dynamics from pure de Sitter exponential expansion. We analyze the models with exponential power-law evolution of the scale factor and we find the equations of influence of non-minimal coupling, choosing it in the special form, on the potential and kinetic energies. Such consideration allows us to substitute the physical potential into obtained equations and then to calculate the non-minimal coupling function and kinetic term that are define GSTG parameters. With this method, we reconstruct GSTG for polynomial, exponential, Higgs, Higgs-Starobinsky and Coleman-Weinberg potentials. Special attention we pay to parameters of cosmological perturbations and prove correspondence obtained solutions to observational data from Planck.