de Sitter limit of inflation and nonlinear perturbation theory
Philip R. Jarnhus, Martin S. Sloth
TL;DR
This paper derives the fourth-order action for the comoving curvature perturbation in inflation using the ADM formalism and a comoving gauge, demonstrating that the de Sitter limit renders the action vanishingly small and the curvature perturbation a pure gauge mode. By performing a nonlinear gauge transformation between the uniform curvature and comoving gauges, it shows how leading slow-roll terms can be reorganized or removed, leading to a final form suppressed by two powers of the slow-roll parameters. The authors extend these insights to conjecture the slow-roll behavior of higher-order $n$-point functions, arguing that odd orders are slow-roll suppressed while even orders may remain unsuppressed, and they provide concrete predictions for higher-point nonlinearity parameters (e.g., $f_{NL}^{(5)}$ and $f_{NL}^{(6)}$ of order $(epsilon^2)$). These results unify the de Sitter limit behavior across gauges, offer a framework for higher-order perturbative calculations, and have implications for loop corrections and non-adiabatic effects in single-field slow-roll inflation.
Abstract
We study the fourth order action of the comoving curvature perturbation in an inflationary universe in order to understand more systematically the de Sitter limit in nonlinear cosmological perturbation theory. We derive the action of the curvature perturbation to fourth order in the comoving gauge, and show that it vanishes sufficiently fast in the de Sitter limit. By studying the de Sitter limit, we then extrapolate to the n'th order action of the comoving curvature perturbation and discuss the slow-roll order of the n-point correlation function.