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

TL;DR

This paper extends the slow-roll formalism for inflation to multi-component inflatons, deriving the curvature perturbation power spectrum to second order in slow-roll. Using the δN formalism and a covariant treatment of field-space geometry, the authors compute how multi-field dynamics and geometry modify the spectrum, spectral index, and its running, reducing to known single-field results in the appropriate limit. The work provides explicit expressions linking the spectrum to the inflaton potential and its derivatives, including terms involving the field-space Riemann tensor and higher-order slow-roll parameters. These results enhance the precision with which multi-field inflationary scenarios can be confronted with Planck-level observations and large-scale structure data.

Abstract

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.