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 ${dn}/{d\ln k}$ to leading order. 
Jin-Ook Gong and Ewan D. Stewart