Growth of scalar-field quantum fluctuations in Robertson-Walker universes

We investigate the behavior of the quantum expectation value of ${\phi }^2$ where φ is a massless, minimally coupled scalar field in a spatially flat Robertson-Walker universe. The scale factor is a(t)∝$t^{\alpha }$, where t is the comoving time. If α≫1, this metric is locally approximately de Sitter space. It is found that 〈${\phi }^2$〉 grows linearly in time for a finite interval during which it is approximated by the de Sitter-space result: 〈${\phi }^2$〉=$H^3$t/(4${\pi }^2$). It subsequently approaches a constant value. If $R_0$ denotes the value of the scalar curvature at the time that growth begins, then the interval of growth lasts for a time of the order of Δt=α(3/$R_0$ ${)}^{1/2}$ and the asymptotic value of 〈${\phi }^2$〉 is α$R_0$/(96${\pi }^2$).