Series formulae for the power spectrum of density perturbations produced during inflation to second order in the general slow-roll approximation

Abstract

Inflation explains why the universe is big, full of matter, and approximately spatially homogeneous, isotropic and flat on the largest observable scales. It also produces curvature perturbations which eventually grow to produce all the structure in the observable universe. These curvature perturbations also provide a unique observational opportunity to determine the more detailed properties of inflation. The standard formulae relating the spectrum of curvature perturbations to the proper-ties of inflation can be incorrect even in the context of slow-roll inflation and current observational bounds. E. D. Stewart provided new formulae, which are robust, to leading order. These formulae will clearly be important if one wants to use observations to probe the properties of inflation in a model independent way. Following the same motivation and formalism, in this paper I calculate the series formula up to second order.