Classicality of Stochastic Noise Away From Quasi-de Sitter Inflation

It is well known that a coarse-grained scalar field living on a de~Sitter (dS) background exhibits classical stochastic behavior, driven by a noise whose amplitude is set by the Hubble constant $H$. The coarse-graining is achieved by discarding wave numbers larger than a cutoff $\sigma aH$ and demanding that $\sigma \ll 1$. Similar results hold for quasi-dS space, where the equation of state parameter $w$ is close to $-1$. Here we present exact expressions for the noise amplitude of a free massless field on an inflationary background with constant $w<-1\mathrm{/}3$. We find that a classical stochastic behavior can emerge for $-5\mathrm{/}3<w<-1\mathrm{/}3$. Furthermore, as we move away from $w=-1$ and approach $w=-1\mathrm{/}3$, the constraint $\sigma \ll 1$ is relaxed and larger cutoffs ($\sigma \sim 1$) become feasible, too. However, in general the amplitude of the noise depends on $\sigma$, except in the quasi-dS regime $w\approx -1$.