Gauge-invariant perturbations at second order: multiple scalar fields on large scales

TL;DR

This work provides a gauge-invariant formulation of second-order perturbations for multiple scalar fields on a flat FRW background, deriving the second-order Klein-Gordon equation in terms of Sasaki-Mukhanov variables on large scales. It expresses the second-order curvature perturbation $\zeta_2$ in terms of first- and second-order field fluctuations, enabling direct computation from the Klein-Gordon dynamics without resorting to slow-roll approximations. The paper includes explicit multi-field expressions for the $0$-$0$ and $0$-$i$ Einstein constraints at second order and demonstrates a slow-roll two-field application that agrees with the Delta-N formalism and lacks non-local contributions. The results facilitate numerical or analytic benchmarking of non-Gaussian signatures in multi-field inflation and pave the way to extending the formalism to smaller scales.

Abstract

We derive the governing equations for multiple scalar fields minimally coupled to gravity in a flat Friedmann-Robertson-Walker (FRW) background spacetime on large scales. We include scalar perturbations up to second order and write the equations in terms of physically transparent gauge-invariant variables at first and second order. This allows us to write the perturbed Klein-Gordon equation at second order solely in terms of the field fluctuations on flat slices at first and second order.