Super-Hubble Nonlinear Perturbations During Inflation

TL;DR

The paper analyzes the nonlinear evolution of long-wavelength perturbations during inflation using a leading-order gradient expansion, introducing a generalized slow-rolling condition and a conserved generalized Bardeen parameter for super-Hubble modes. It shows that single-field reheating does not amplify these modes, re-derives the stochastic inflation equation including gravitational fluctuations, and demonstrates that vector perturbations arise when slow-rolling fails. The results clarify back-reaction issues, show the local Friedmann equations remain valid for adiabatic infrared modes, and reveal that nonlinearities can generate vector modes in certain regimes. Together, these findings refine the understanding of nonlinear effects and stochastic dynamics in inflationary cosmology and their observational implications.

Abstract

We show that the non-linear evolution of long wavelength perturbations may be important in a wide class of inflationary scenarios. We develop a solution for the evolution of such nonlinear perturbations which is exact to first order in a gradient expansion. As a first application, we demonstrate that in single field models of inflation there can be no parametric amplification of super-Hubble modes during reheating. We consider the implications of the solution for recent discussions of the back-reaction effect of long wavelength perturbations on the background geometry, give a new derivation of the equation of motion of stochastic inflation, and demonstrate that if the (generalized) slow-rolling condition is not satisfied, then inevitably long wavelength vector modes for gravitational fluctuations will be generated.

Figures (1)

• Figure 1: The solid curve is a solution for $g(\varphi)$ during the reheating era and the dashed curve is $\sqrt{V(\varphi)/(6 \pi G) }$. The horizontal axis is in units of Planck mass while the vertical axis has arbitrary units.