Extended δN Formalism: Nonspatially Flat Separate-Universe Approach

The δN formalism is a powerful approach to compute nonlinearly the large-scale evolution of the comoving curvature perturbation ζ. It assumes a set of FLRW patches that evolve independently, but in doing so, all the gradient terms are discarded, which are not negligibly small in models beyond slow roll. In this Letter, we extend the formalism to capture these gradient corrections by encoding them in a homogeneous-spatial-curvature contribution assigned to each FLRW patch. For a concrete example, we apply this formalism to the ultra-slow-roll inflation, and find that it can correctly describe the large-scale evolution of the comoving curvature perturbation from the horizon exit. We also briefly discuss non-Gaussianities in this context.