Quantum probability distributions in the early Universe. IV. Stochastic dynamics in de Sitter space

Using the Smoluchowski equation, we investigate the stochastic evolution of the horizon-averaged or coarse-grained scalar field (inflaton) in a pure de Sitter background. We clarify the effect quantum fluctuations have on the classical dynamics of relaxation. We consider two types of nonlinear potentials [V(φ)=(1/2γ${\phi }^2$+ 1) / 4 g${\phi }^4$ and V(φ)=-(1/2λ${\phi }^2$+ 1) / 4 g${\phi }^4$] with inflaton probability distributions initially displaced from equilibrium. In the former case quantum fluctuations have only a minor effect on the classical inflaton dynamics. In the latter case the situation is different. Quantum fluctuations play a crucial part in the early-time behavior of the inflaton probability distribution. Using techniques borrowed from nonequilibrium statistical mechanics, we show how [for V(φ)=-(1/2λ${\mathrm{\phi }}^2$+ 1) / 4 g${\mathrm{\phi }}^4$] macroscopic (classical) order originates from stochastic (quantum) initial conditions. We estimate the time scale at which this transition takes place. The work here extends and validates the conclusion of Guth and Pi that the long-time behavior of f(φ;t) can be described by a classical probability distribution.