Free scalar correlators in de Sitter via the stochastic approach beyond slow roll

TL;DR

This paper extends the stochastic approach to scalar correlators in de Sitter space beyond slow roll, demonstrating its accuracy for free fields and discussing challenges for interacting theories.

Contribution

It shows how to correctly implement the stochastic approach beyond slow roll and highlights issues with naive horizon cut-offs.

Findings

Stochastic approach reproduces exact long-distance correlators for free fields.

Naive horizon cut-off does not yield correct results without specific adjustments.

Implications for interacting theories are discussed.

Abstract

The stochastic approach to calculating scalar correlation functions in de Sitter spacetime is extended beyond the overdamped "slow roll" approximation. We show that with the correct noise term, it reproduces the exact asymptotic long-distance behaviour of field correlators in free field theory, thereby demonstrating the viability of the technique. However, we also show that the na\"ive way of calculating the noise term by introducing a cut-off at the horizon does not give the correct answer unless the cut-off is chosen specifically to give the required result. We discuss the implications of this for interacting theories.