Adaptive asymptotic solutions of inflationary models in the Hamilton-Jacobi formalism: Application to T-models

Abstract

We develop a method to compute the slow-roll expansion for the Hubble parameter in inflationary models in a flat Friedmann-Lema\^itre-Robertson-Walker spacetime that is applicable to a wide class of potentials including monomial, polynomial, or rational functions of the inflaton, as well as polynomial or rational functions of the exponential of the inflaton. The method, formulated within the Hamilton-Jacobi formalism, adapts the form of the slow-roll expansion to the analytic form of the inflationary potential, thus allowing a consistent order-by-order computation amenable to Pad\'e summation. Using T-models as an example, we show that Pad\'e summation extends the domain of validity of this adapted slow-roll expansion to the end of inflation. Likewise, Pad\'e summation extends the domain of validity of kinetic-dominance asymptotic expansions of the Hubble parameter into the fast-roll regime, where they can be matched to the aforesaid Pad\'e-summed slow-roll expansions. This matching in turn determines the relation between the expansions for the number $N$ of e-folds and allows us to compute the total amount of inflation as a function of the initial data or, conversely, to select initial data that correspond to a fixed total amount of inflation. Using the slow-roll stage expansions, we also derive expansions for the corresponding spectral index $n_s$ accurate to order $1/N^2$, and tensor-to-scalar ratio $r$ accurate to order $1/N^3$ for these T-models.