Langevin Analysis of Eternal Inflation

TL;DR

This paper analyzes the conditions under which eternal inflation occurs using a Langevin approach on a quartic potential, showing that typical stochastic effects do not lead to eternal inflation unless specific selection effects are considered.

Contribution

It provides an analytic investigation of eternal inflation in a quartic potential model, highlighting the impact of different weighting and selection effects on inflation's eternity.

Findings

Stochastic noise has little impact on total inflationary efolds without weighting.

Volume weighting alone does not produce eternal inflation.

Selection effects can induce a scenario similar to eternal inflation.

Abstract

It has been widely claimed that inflation is generically eternal to the future, even in models where the inflaton potential monotonically increases away from its minimum. The idea is that quantum fluctuations allow the field to jump uphill, thereby continually revitalizing the inflationary process in some regions. In this paper we investigate a simple model of this process, pertaining to inflation with a quartic potential, in which analytic progress may be made. We calculate several quantities of interest, such as the expected number of inflationary efolds, first without and then with various selection effects. With no additional weighting, the stochastic noise has little impact on the total number of inflationary efoldings even if the inflaton starts with a Planckian energy density. A "rolling" volume factor, i.e. weighting in proportion to the volume at that time, also leads to a monotonically decreasing Hubble constant and hence no eternal inflation. We show how stronger selection effects including a constraint on the initial and final states and weighting with the final volume factor can lead to a picture similar to that usually associated with eternal inflation.