New method of exponential potentials reconstruction based on given scale factor in phantonical two-field models

We investigate two-field cosmological model with phantom and canonical fields (phantonical model as a generalisation of the quintom model for global universe evolution, including early inflationary stage). The model is represented as the chiral cosmological model with the target space conformal to 2D pseudo-Euclidean space. We found three sorts of exact solutions for a constant potential by direct integration of dynamic equations and proposed new method of exact solution construction also extended for e-folds N-formalism for the case of non-constant exponential potential. We show that the exact solutions of cosmological dynamic equations can be obtained in explicit form for any type of scale factor evolution $a(t)$ which implies the explicit inverse dependence $t=t(a)$, considering the quasi de Sitter expansion of the universe with non-negligible kinetic energies of scalar fields and showing that the appeared effective cosmological constant can be considered as the source of second accelerated expansion of the universe. Further we analyze cosmological perturbations in the two-field model under consideration reducing it to the single field one. Such transition give us the way of cosmological parameters calculation and comparison them to observational data. We find that in proposed two-field cosmological model the isocurvature perturbations are negligible, and observable curvature perturbations are induced by adiabatic modes only. The series of phantonical models based on exact inflationary solutions are represented, and it is shown the correspondence to observational data for these models.