Late time solution for interacting scalar in accelerating spaces

We consider stochastic inflation in an interacting scalar field in spatially homogeneous accelerating space-times with a constant principal slow roll parameter $ε$. 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) $ρ$ which is a function of $φ/H$ only, where $φ=φ(\vec x)$ is the scalar field and $H=H(t)$ denotes the Hubble parameter. We give explicit late-time solutions for $ρ\rightarrow ρ_\infty(φ/H)$, and thereby find the order $ε$ 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 $ε=$ constant.

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