Formalising the Slow-Roll Approximation in Inflation

TL;DR

The paper formalises the slow-roll approximation in inflation by rigorously contrasting the potential-based PSRA with the Hubble-based HSRA, and by highlighting the central role of the inflationary attractor. It develops a slow-roll expansion built on an infinite yet effectively finite-at- each-order hierarchy of parameters for $H(\phi)$ and $V(\phi)$, enabling analytic progress beyond leading order. It introduces a refined inflation measure, barN, based on the contraction of the comoving Hubble length, and shows how to compute observables with higher-order corrections. To extend the validity of the expansion, the authors adopt rational approximants (Canterbury, including Padé-like forms) and demonstrate concrete two- and three-variable implementations, including explicit expressions for the φ^2 potential. The work provides a practical framework for accurate inflationary dynamics calculations across a broad class of models and clarifies the interplay between attractor behavior, higher-order corrections, and perturbation predictions.

Abstract

The meaning of the inflationary slow-roll approximation is formalised. Comparisons are made between an approach based on the Hamilton-Jacobi equations, governing the evolution of the Hubble parameter, and the usual scenario based on the evolution of the potential energy density. The vital role of the inflationary attractor solution is emphasised, and some of its properties described. We propose a new measure of inflation, based upon contraction of the comoving Hubble length as opposed to the usual e-foldings of physical expansion, and derive relevant formulae. We introduce an infinite hierarchy of slow-roll parameters, and show that only a finite number of them are required to produce results to a given order. The extension of the slow-roll approximation into an analytic slow-roll expansion, converging on the exact solution, is provided. Its role in calculations of inflationary dynamics is discussed. We explore rational-approximants as a method of extending the range of convergence of the slow-roll expansion up to, and beyond, the end of inflation.