Path Integral for Stochastic Inflation: Non-Perturbative Volume Weighting, Complex Histories, Initial Conditions and the End of Inflation

TL;DR

This paper develops a path integral approach to stochastic inflation, revealing how volume weighting influences inflation's evolution, initial condition independence, and the end via slow-roll, with complex histories playing a role.

Contribution

It introduces a non-perturbative path integral formulation of stochastic inflation that incorporates volume weighting and explores complex histories and initial conditions.

Findings

Volume weighting can be implemented in the path integral framework.

Inflation loses memory of initial conditions under volume weighting.

Inflation typically ends via slow-roll, mitigating Youngness Paradox criticisms.

Abstract

In this paper we present a path integral formulation of stochastic inflation, in which volume weighting can easily be implemented. With an in-depth study of inflation in a quartic potential, we investigate how the inflaton evolves and how inflation typically ends both with and without volume weighting. Perhaps unexpectedly, complex histories sometimes emerge with volume weighting. The reward for this excursion into the complex plane is an insight into how volume-weighted inflation both loses memory of initial conditions and ends via slow-roll. The slow-roll end of inflation mitigates certain "Youngness Paradox"-type criticisms of the volume-weighted paradigm. Thus it is perhaps time to rehabilitate proper time volume weighting as a viable measure for answering at least some interesting cosmological questions.