Stochastic Samples versus Vacuum Expectation Values in Cosmology

TL;DR

This paper compares stochastic and expectation value approaches in cosmology, showing that for certain operators like the scalar field squared, expectation values are reliable, and highlights the complementary roles of both methods in understanding quantum back-reaction during inflation.

Contribution

It demonstrates that expectation values can be valid for specific operators in cosmology and clarifies the relationship between stochastic formalism and expectation values in quantum back-reaction analysis.

Findings

Expectation values are reliable for the scalar field squared.

Stochastic formalism captures leading infrared logarithm corrections.

Expectation values and stochastic methods are complementary.

Abstract

Particle theorists typically use expectation values to study the quantum back-reaction on inflation, whereas many cosmologists stress the stochastic nature of the process. While expectation values certainly give misleading results for some things, such as the stress tensor, we argue that operators exist for which there is no essential problem. We quantify this by examining the stochastic properties of a noninteracting, massless, minimally coupled scalar on a locally de Sitter background. The square of the stochastic realization of this field seems to provide an example of great relevance for which expectation values are not misleading. We also examine the frequently expressed concern that significant back-reaction from expectation values necessarily implies large stochastic fluctuations between nearby spatial points. Rather than viewing the stochastic formalism in opposition to expectation values, we argue that it provides a marvelously simple way of capturing the leading infrared logarithm corrections to the latter, as advocated by Starobinsky.