Non-Gaussianity of superhorizon curvature perturbations beyond δ N formalism

Abstract

We develop a theory of nonlinear cosmological perturbations onsuperhorizon scales for a single scalar field with a general kinetic term and a general form of the potential. We employ the ADM formalism and the spatial gradient expansion approach, characterised by O(ϵm), where ϵ = 1/(HL) is a small parameter representing the ratio of the Hubble radius to the characteristic length scale L of perturbations. We obtain the general solution for a full nonlinear versionof the curvature perturbation valid up through second-order in ϵ (m = 2). We find the solution satisfies a nonlinear second-order differentialequation as an extension of the equation for the linear curvatureperturbation on the comoving hypersurface.Then we formulate a general method to match a perturbative solution accurate to n-th-order in perturbation inside the horizon to our nonlinear solution accurate to second-order (m = 2) inthe gradient expansion on scales slightly greater than the Hubble radius.The formalism developed in this paper allows us to calculate the superhorizon evolution of a primordial non-Gaussianity beyond the so-called δNformalism or separate universe approach which is equivalentto leading order (m = 0) in the gradient expansion.In particular, it can deal with the case when there isa temporary violation of slow-roll conditions.As an application of our formalism, we consider Starobinsky's model, which is a single field model having a temporary non-slow-roll stage due to a sharp change inthe potential slope. We find that a large non-Gaussianity can be generated even on superhorizonscales due to this temporary suspension of slow-roll inflation.