Friction in Stochastic Inflation

TL;DR

This paper derives an exact probability distribution for curvature fluctuations in stochastic inflation with friction, revealing exponential tails and differences between forward and reverse approaches, especially in eternal inflation scenarios.

Contribution

It introduces a time-reversed stochastic inflation framework with friction, providing exact solutions and comparing them to traditional methods, highlighting new insights into inflationary fluctuations.

Findings

Exponential decay in curvature fluctuation tails contrasts with Levy-like power laws.

Time-reversed formalism yields Gaussian tails in the classical limit.

Negative drift leads to eternal inflation with finite curvature distribution.

Abstract

We solve time-reversed stochastic inflation in the semi-infinite flat potential with a constant drift term and derive an exact expression for the probability distribution of the curvature fluctuations. It exhibits exponential decaying tails which contrast to the Levy-like power law behaviour encountered without friction. Such a non-vanishing drift acts as a regulator for the conventional ``forward'' stochastic $\delta N$-formalism, which is otherwise ill-defined in the unbounded and flat potentials typical of plateau models of inflation. This setup therefore allows us to compare the curvature distribution derived from both approaches, reverse and forward in time. Up to similar exponential tails, we find quantitative differences. In particular, in the classical-like limit of very large drift, the tails become Gaussian but only in the time-reversed picture. As a toy model of eternal inflation, we finally discuss the case of negative drift in which inflation never ends for many field trajectories. The forward approach becomes pathological whereas the reverse formalism gives back a finite curvature distribution with always exponential tails. All these differences end up being related to the very definition of the background which is ambiguous when a classical trajectory does not exist.