New perturbative method for analytical solutions in single-field models of inflation

Abstract

We propose a new parametrization of the background equations of motion corresponding to (canonical) single-field models of inflation, which allows a better understanding of the general properties of the solutions and of the corresponding predictions in the inflationary observables. Based on the tools of dynamical systems, the method suggests that inflation comes in two flavors: power-law and de Sitter. Power-law inflation seems to occur for a restricted type of potentials, whereas de Sitter inflation has a much broader applicability. We also show a general perturbative method, by means of series expansion, to solve the new equations of motion around the critical point of the de Sitter type, and how the method can be used for arbitrary models of de Sitter inflation. It is then argued that for the latter there are two general classes of inflationary solutions, given in terms of the behavior of the tensor-to-scalar ratio as a function of the number $N$ of $e$-folds before the end of inflation: $r \sim N^{-1}$ (Class I), or $r \sim N^{-2}$ (Class II). We give some examples of the two classes in terms of known scalar field potentials, and compare their general predictions with constraints obtained from observations.