Statistical nature of infrared dynamics on de Sitter background

TL;DR

This paper develops a systematic method to derive effective equations of motion for long-wavelength scalar fields in de Sitter space, extending stochastic formalism to include sub-leading secular growth, and confirms the classical stochastic nature of the dynamics.

Contribution

It introduces an extended stochastic formalism that captures secular growth in scalar fields on de Sitter background, including sub-leading effects, and applies it to $bbb4$ theory.

Findings

Effective EoM reproduces next-to-leading secular growth.

The extended formalism describes all secularly growing correlation functions.

The effective dynamics can be interpreted as a classical stochastic process.

Abstract

In this study, we formulate a systematic way of deriving an effective equation of motion(EoM) for long wavelength modes of a massless scalar field with a general potential $V(\phi)$ on de Sitter background, and investigate whether or not the effective EoM can be described as a classical stochastic process. Our formulation gives an extension of the usual stochastic formalism to including sub-leading secular growth coming from the nonlinearity of short wavelength modes. Applying our formalism to $\lambda \phi^4$ theory, we explicitly derive an effective EoM which correctly recovers the next-to-leading secularly growing part at a late time, and show that this effective EoM can be seen as a classical stochastic process. Our extended stochastic formalism can describe all secularly growing terms which appear in all correlation functions with a specific operator ordering, which will not be a big drawback because the commutator of a light scalar field becomes negligible at large scales owing to the squeezing.