Late time solution for interacting scalar in accelerating spaces

TL;DR

This paper derives late-time probability distributions for an interacting scalar field in accelerating universes, including corrections for slow roll parameters, enabling calculation of scalar field correlations in such spacetimes.

Contribution

It provides explicit late-time solutions for the scalar field PDF in accelerating spaces with scale-invariant potentials, including first-order epsilon corrections to previous models.

Findings

Derived late-time probability distribution functions for scalar fields.

Calculated epsilon-order corrections to the Starobinsky-Yokoyama result.

Enabled computation of n-point functions in accelerating universes.

Abstract

We consider stochastic inflation in an interacting scalar field in spatially homogeneous accelerating space-times with a constant principal slow roll parameter $\epsilon$. We show that, if the scalar potential is scale invariant (which is the case when scalar contains quartic self-interaction and couples non-minimally to gravity), the late-time solution on accelerating FLRW spaces can be described by a probability distribution function (PDF) $\rho$ which is a function of $\varphi/H$ only, where $\varphi=\varphi(\vec x)$ is the scalar field and $H=H(t)$ denotes the Hubble parameter. We give explicit late-time solutions for $\rho\rightarrow \rho_\infty(\varphi/H)$, and thereby find the order $\epsilon$ corrections to the Starobinsky-Yokoyama result. This PDF can then be used to calculate e.g. various $n-$point functions of the (self-interacting) scalar field, which are valid at late times in arbitrary accelerating space-times with $\epsilon=$ constant.