The construction of exact solutions in scalar field inflationary cosmology is of growing interest. In this work, we review the results which have been obtained with the help of one of the most effective methods, viz., the method of generating functions for the construction of exact solutions in scalar field cosmology. We also include in the debate the superpotential method, which may be considered as the bridge to the slow roll approximation equations. Based on the review, we suggest a classification for the generating functions, and find a connection for all of them with the superpotential. PACS numbers: 98.80. Bp, 98.80.Cq, 98.80.Es, 98.80.Jk I. EARLY INFLATION AND THE IMPLEMENTED SCALAR FIELD

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We describe how to reconstruct generalized scalar-tensor gravity (GSTG) theory, which admits exact solutions for a physical type of 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 an a priori specified relation between the Hubble parameter H and a function of a non-minimal coupling F = 1 + δ F, H ∝ √ F. Deviations from Einstein gravity δ F induce corresponding deviations of the potential δV from a constant value and modify the dynamics from a 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 a consideration allows us to substitute the physical potential into the obtained equations and then to calculate the non-minimal coupling function and kinetic term that define the GSTG parameters. With this method, we reconstruct GSTG for the polynomial, exponential, Higgs, Higgs-Starobinsky and Coleman-Weinberg potentials. Special attention we pay to parameters of cosmological perturbations and prove the correspondence of the obtained solutions to observational data from Planck.

We study the special class of the exact solutions in cosmological models based on the Generalized Scalar-Tensor Gravity with non-minimal coupling of a scalar field to the Ricci scalar and to the Gauss-Bonnet scalar in 4D Friedmann universe corresponding to similar ones in GR. The parameters of cosmological perturbations in such models correspond to the case of Einstein gravity with a high precision. As the example of proposed approach, we obtain the exact solutions for the power-law and exponential power-law inflation. * ingvor@inbox.ru † chervon.sergey@gmail.com 1 We have to stress that Starobinsky model [1] was related to modified f(R) gravity.