A Stochastic Measure for Eternal Inflation

TL;DR

This paper applies a stochastic approach to measure the likelihood of different Hubble radii in eternal inflation, revealing classical dominance in certain transitions and entropy suppression in others.

Contribution

It introduces a stochastic framework for calculating probabilities in eternal inflation, highlighting the role of de Sitter entropy in transition likelihoods.

Findings

Classical probability dominates from smaller to larger Hubble radius.

Transition from larger to smaller Hubble radius is suppressed by entropy.

The stochastic approach naturally incorporates de Sitter entropy.

Abstract

We use the stochastic approach to investigate the measure for slow roll eternal inflation. The probability for the universe of a given Hubble radius can be calculated in this framework. In a solvable model, it is shown that the probability for the universe to evolve from a state with a smaller Hubble radius to that of a larger Hubble radius is dominated by the classical probability without the stochastic source. While the probability for the universe to evolve from a larger Hubble radius to a smaller one is suppressed by $\exp(-\Delta S)$, where the de Sitter entropy $S$ arises naturally in this stochastic approach.