Stochastic inflation in the Constant Roll regime

Abstract

We investigate the inhomogeneities generated during the inflationary epoch from the point of view of the stochastic formalism, which attempts to transform a problem of quantum fluctuations into a statistical one. The formalism, that we derive in the text, is based on the use of the Arnowitt-Deser-Misner (ADM) equations, which are convenient to describe inhomogeneities in the context of inflation, as well as gradient expansion, which works at zeroth order in spatial gradients but at all orders in the amplitudes of the fluctuations, and is therefore intended to capture non-perturbative effects. Finally, the perturbations are split into long- and short-wavelength modes, where the latter act as a stochastic noise for the former when crossing a certain scale. We demonstrate that the use of certain approximations in the derivation of this formalism, which are intended to make the system of stochastic differential equations (SDEs) Markovian and described with white noises, causes the method to become restricted to the reproduction of Linear Perturbation Theory (LPT). This framework, nonetheless, is still useful since it can be used as a test for the validity of the linear approximation, signalling the coming into play of non-perturbative effects. Specifically, we solve the system of SDEs numerically for the Constant Roll (CR) inflationary scenario, and show that this regime is in accordance with LPT.