Statistical nature of infrared dynamics on de Sitter background

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 $V(\varphi )$ 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 $\lambda {\varphi }^4$ 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.

Comments: 20 pages plus appendices, 12 figures. v4: matches published version