The Spectrum of Density Perturbations produced during Inflation to Leading Order in a General Slow-Roll Approximation

The standard calculation of the spectrum of density perturbations produced during inflation assumes (i) |n-1| << 1 and (ii) |dn/dlnk| << |n-1|. Slow-roll predicts and observations require (i), but neither slow-roll nor observations require (ii). In this paper I derive formulae for the spectrum, spectral index, and running of the spectral index, assuming only (i) and not (ii). I give a large class of observationally viable examples in which these general slow-roll formulae are accurate but the standard slow-roll formulae for the spectral index and running are incorrect.