A short note on the curvature perturbation at second order

Working with perturbations about a Friedmann-Lemaitre-Robertson-Walker 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.