A not so short note on the Klein-Gordon equation at second order

TL;DR

This work derives the Klein-Gordon equation at second order for a system of multiple scalar fields in a flat FRW background, including metric perturbations up to second order in the flat gauge. It provides a closed-form real-space expression and its Fourier-space counterpart, showing that the second-order equation contains finite, gradient-based source terms and remains well-behaved on large scales without infrared divergences. The authors also develop a slow-roll approximation to substantially simplify the second-order equation, yielding explicit expressions for the mass matrix and slow-roll parameters that facilitate practical computations. These results enable efficient calculations of non-Gaussian observables by linking field fluctuations to curvature perturbations and related quantities like $f_{ m NL}$, with potential extensions to vector perturbations and numerical implementations.

Abstract

We give the governing equations for multiple scalar fields in a flat Friedmann-Robertson-Walker (FRW) background spacetime on all scales, allowing for metric and field perturbations up to second order. We then derive the Klein-Gordon equation at second order in closed form in terms of gauge-invariant perturbations of the fields in the uniform curvature gauge. We also give a simplified form of the Klein-Gordon equation using the slow-roll approximation.