Renormalisation and matching of massless scalar correlation functions in Soft de Sitter Effective Theory

TL;DR

This paper develops a Soft de Sitter Effective Theory (SdSET) for massless scalar fields, demonstrating its renormalisation and matching procedures, and validating its effectiveness through explicit calculations of correlation functions.

Contribution

It constructs SdSET in dimensional regularisation, including initial conditions, and shows how to perform renormalisation and matching for superhorizon modes in de Sitter space.

Findings

Successfully matches trispectrum and six-point function at tree level.

Verifies one-loop power spectrum matches between UV theory and SdSET.

Confirms SdSET as a valid effective field theory for superhorizon scalar dynamics.

Abstract

For light and massless scalar fields, cosmological correlation functions suffer from infrared divergences and secular logarithms. Soft de Sitter Effective Theory (SdSET) has been proposed by Cohen and Green as the effective description of the non-trivial dynamics of long-wavelength modes $k_{\rm phys} < H$ in de Sitter space, which is responsible for the infrared and late-time logarithms, and as a systematic extension of the stochastic approach. In this article, we construct SdSET in dimensional regularisation, including an initial-condition functional. We demonstrate by examples that renormalisation and matching works as for flat-space effective field theories. Adopting massless $\kappa \phi^4$ theory as the UV theory, we match the tree-level trispectrum and six-point function, and the one-loop power spectrum to SdSET, verifying explicitly that SdSET is the appropriate effective field theory for the quantum dynamics of superhorizon modes.