The Power Spectrum for a Multi-Component Inflaton to Second-Order Corrections in the Slow-Roll Expansion

We derive the power spectrum $\mathcal{P}(k)$ of the density perturbations produced during inflation up to second-order corrections in the standard slow-roll approximation for an inflaton with more than one degree of freedom. We also present the spectral index $n$ up to first-order corrections including previously missing terms, and the running $dn\mathrm{/}d\ln k$ to leading order.