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

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