A short note on the curvature perturbation at second order

TL;DR

This paper derives a gauge-invariant expression for second-order curvature perturbations in FLRW spacetime using scalar field fluctuations, facilitating future numerical studies of cosmological perturbations.

Contribution

It provides a novel second-order gauge-invariant formula for curvature perturbations solely in terms of scalar field fluctuations, applicable to both large and small scales.

Findings

Derived second-order gauge-invariant curvature perturbation expression.

Expressed results in terms of scalar field fluctuations and derivatives.

Facilitates numerical implementation of second-order cosmological perturbations.

Abstract

Working with perturbations about an FLRW spacetime, we compute the gauge-invariant curvature perturbation to second order solely in terms of scalar field fluctuations. Using the curvature perturbation on uniform density hypersurfaces as our starting point, we give our results in terms of field fluctuations in the flat gauge, incorporating both large and small scale behaviour. For ease of future numerical implementation we give our result in terms of the scalar field fluctuations and their time derivatives.