Spectator field dynamics in de Sitter and curvaton initial conditions

We investigate the stochastic behaviour of long wavelength modes of light spectator scalar fields during inflation. When starting from a classical field value, the probability distribution for the spectator both spreads out and moves towards an equilibrium distribution. We study the timescales for a mixed quadratic and quartic potential. The timescale of equilibration depends on the parameters of the model, and can be surprisingly large, more than thousands of e-folds for the majority of light spectators. These results imply that the initial conditions for spectator fields are not automatically erased during inflation. These general results can be used to determine the probability of the value of the zero-mode inside a Universe-sized patch, which is relevant for observations. As an example, we apply the results to the curvaton model to calculate the probability distribution of the curvature perturbation and discuss 'typical' Universes