Gauge-invariant perturbations at second order in two-field inflation

We study the second-order gauge-invariant adiabatic and isocurvature perturbations in terms of the scalar fields present during inflation, along with the related fully non-linear space gradient of these quantities. We discuss the relation with other perturbation quantities defined in the literature. We also construct the exact cubic action of the second-order perturbations (beyond any slow-roll or super-horizon approximations and including tensor perturbations), both in the uniform energy density gauge and the flat gauge in order to settle various gauge-related issues. We thus provide the tool to calculate the exact non-Gaussianity beyond slow-roll and at any scale.