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Full nonlinear growing and decaying modes of superhorizon curvature perturbations

We clarify the behavior of curvature perturbations in a nonlinear theory in case the inflaton temporarily stops during inflation. We focus on the evolution of curvature perturbation on superhorizon scales by adopting the spatial gradient expansion and show that the nonlinear theory, called the {\it beyond} $δN$-formalism for a general single scalar field as the next-leading order in the expansion. Both the leading-order in the expansion ($δN$-formalism) and our nonlinear theory include the solutions of full-nonlinear orders in the standard perturbative expansion. Additionally, in our formalism, we can deal with the time evolution in contrast to $δN$-formalism, where curvature perturbations remain just constant, and show decaying modes do not couple with growing modes as similar to the case with linear theory. We can conclude that although the decaying mode diverges when $\dotϕ$ vanishes, there appears no trouble for both the linear and nonlinear theory since these modes will vanish at late times.