Late time solution for interacting scalar in accelerating spaces

Understand

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.