Formalizing the slow-roll approximation in inflation

The meaning of the inflationary slow-roll approximation is formalized. 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 emphasized, 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 formulas. 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.