Stochastic Eternal Inflation in a Bianchi Type I Universe

TL;DR

This paper investigates the conditions under which stochastic eternal inflation can occur in a Bianchi Type I universe, emphasizing the role of anisotropy and deriving probabilistic criteria using stochastic calculus.

Contribution

It introduces a stochastic Klein-Gordon equation for Bianchi Type I spacetime and identifies the small anisotropy conditions necessary for eternal inflation.

Findings

Eternal inflation requires small anisotropy levels.

Probability of eternal inflation depends on shear anisotropy variables.

Eternal inflation occurs within a specific region of the Kasner circle.

Abstract

The phenomenon of stochastic eternal inflation is studied for a chaotic inflation potential in a Bianchi Type I spacetime background. After deriving the appropriate stochastic Klein-Gordon equation, we give details on the conditions for eternal inflation. It is shown that for eternal inflation to occur, the amount of anisotropy must be small. In fact, it is shown that eternal inflation will only take place if the shear anisotropy variables take on values within a small region of the interior of the Kasner circle. We then calculate the probability of eternal inflation occurring based on techniques from stochastic calculus.