  We study the stochastic distribution of spectator fields predicted in
different slow-roll inflation backgrounds. Spectator fields have a negligible
energy density during inflation but may play an important dynamical role later,
even giving rise to primordial density perturbations within our observational
horizon today. During de-Sitter expansion there is an equilibrium solution for
the spectator field which is often used to estimate the stochastic distribution
during slow-roll inflation. However slow roll only requires that the Hubble
rate varies slowly compared to the Hubble time, while the time taken for the
stochastic distribution to evolve to the de-Sitter equilibrium solution can be
much longer than a Hubble time. We study both chaotic (monomial) and plateau
inflaton potentials, with quadratic, quartic and axionic spectator fields. We
give an adiabaticity condition for the spectator field distribution to relax to
the de-Sitter equilibrium, and find that the de-Sitter approximation is never a
reliable estimate for the typical distribution at the end of inflation for a
quadratic spectator during monomial inflation. The existence of an adiabatic
regime at early times can erase the dependence on initial conditions of the
final distribution of field values. In these cases, spectator fields acquire
sub-Planckian expectation values. Otherwise spectator fields may acquire much
larger field displacements than suggested by the de-Sitter equilibrium
solution. We quantify the information about initial conditions that can be
obtained from the final field distribution. Our results may have important
consequences for the viability of spectator models for the origin of structure,
such as the simplest curvaton models.
