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 formulas for the spectrum P, spectral index $n,$ and running of the spectral index dn/dlnk, assuming only (i) and not (ii). I give a large class of observationally viable examples in which these general slow-roll formulas are accurate but the standard slow-roll formulas for the spectral index and running are incorrect.