Lagrangian Formulation of Stochastic Inflation: A Recursive Approach

Abstract

We present a new, recursive approach to stochastic inflation which is self-consistent and solves multiple problems which plagued a certain number of previous studies, in particular in realistic contexts where the background spacetime is taken to be dynamical, where there is more than one field present, especially with a mass hierarchy, or where the role played by back-reaction is suspected to be important. We first review the formalism of stochastic inflation as it is usually heuristically presented, that is, deriving the Langevin equations from the field equations of motion, and summarize previous results on the subject. We demonstrate where inconsistent approximations to the Langevin equations are commonly made, and show how these can be avoided. This set up shares many similarities with quantum Brownian motion and out-of-equilibrium statistical quantum dynamics. We hence review how path integral techniques can be applied to the stochastic inflationary context. We show this formalism to be consistent with the standard approach, develop a natural perturbative expansion, and use it to calculate the one-loop corrected Langevin equations.