Beyond N formalism

Abstract

We develop a theory of nonlinear cosmological perturbations on superhorizon scales for a multi-component scalar field with a general kinetic term and a general form of the potential in the context of inflationary cosmology. We employ the ADM formalism and the spatial gradient expansion approach, characterized by ${\mathcal {O}}(\epsilon ^2)$, where ϵ = 1/(HL) is a small parameter representing the ratio of the Hubble radius to the characteristic length scale L of perturbations. We provide a formalism to obtain the solution in the multi-field case. This formalism can be applied to the superhorizon evolution of a primordial non-Gaussianity beyond the so-called δN formalism which is equivalent to ${\mathcal {O}}(\epsilon ^0)$ of the gradient expansion. In doing so, we also derive fully nonlinear gauge transformation rules valid through ${\mathcal {O}}(\epsilon ^2)$. These fully nonlinear gauge transformation rules can be used to derive the solution in a desired gauge from the one in a gauge where computations are much simpler. As a demonstration, we consider an analytically solvable model and construct the solution explicitly.