Stochastic approach to de Sitter instability and eternal inflation

TL;DR

This paper uses a stochastic formalism to analyze the stability of de Sitter space and the conditions for eternal inflation, deriving bounds that relate to the de Sitter entropy limit.

Contribution

It introduces a novel stochastic approach that derives bounds on de Sitter stability and eternal inflation, connecting these bounds to the de Sitter entropy limit.

Findings

Derived a Fokker-Planck equation without second order time derivatives.

Obtained bounds for non-eternal inflation and de Sitter background stability.

Connected the bounds to the de Sitter entropy limit.

Abstract

We investigate when effective theories of a scalar field on (quasi-)de Sitter background break down through the stochastic formalism. We derive the Fokker-Planck equation leaving the second order time derivative of the scalar field. Assuming there exists an equilibrium distribution for the field velocity, we obtain a mean value and a variance of the field velocity caused by the quantum fluctuation. Introducing coarse-grained Einstein equations, we obtain bounds for the non-eternal inflation phase and for maintaining the exact de Sitter background. We point out that those bounds derived in our formalism correspond to the de Sitter entropy bound proposed by Arkani-Hamed, \textit{et.al.}, up to $O(1)$ factor, even for a massless free scalar field on exact de Sitter background. We discuss connections of our results to the quantum field theory also.