Particle production and classical condensates in de Sitter space

The cosmological particle production in a $k=0$ expanding de Sitter universe with a Hubble parameter $H_0$ is considered for various values of mass or conformal coupling of a free, scalar field. One finds that, for a minimally coupled field with mass $0 \leq m^2 < 9 H_0^2/4$ (except for $m^2= 2H_0^2$), the one-mode occupation number grows to unity soon after the physical wavelength of the mode becomes larger than the Hubble radius, and afterwards diverges as $n(t) \sim O(1)(\lambda_{phys}(t)/H_0^{-1})^{2\nu}$, where $\nu \equiv [9/4 - m^2/H_0^2]^{1/2}$. However, for a field with $m^2 > 9H_0^2/4$, the occupation number of a mode outside the Hubble radius is rapidly oscillating and bounded and does not exceed unity. These results, readily generalized for cases of a nonminimal coupling, provide a clear argument that the long-wavelength vacuum fluctuations of low-mass fields in an inflationary universe do show classical behavior, while those of heavy fields do not. The interaction or self-interaction does not appear necessary for the emergence of classical features, which are entirely due to the rapid expansion of the de Sitter background and the upside-down nature of quantum oscillators for modes outside the Hubble radius.