The stochastic spectator

Abstract

We study the stochastic distribution of spectator fields predicted in\ndifferent slow-roll inflation backgrounds. Spectator fields have a negligible\nenergy density during inflation but may play an important dynamical role later,\neven giving rise to primordial density perturbations within our observational\nhorizon today. During de-Sitter expansion there is an equilibrium solution for\nthe spectator field which is often used to estimate the stochastic distribution\nduring slow-roll inflation. However slow roll only requires that the Hubble\nrate varies slowly compared to the Hubble time, while the time taken for the\nstochastic distribution to evolve to the de-Sitter equilibrium solution can be\nmuch longer than a Hubble time. We study both chaotic (monomial) and plateau\ninflaton potentials, with quadratic, quartic and axionic spectator fields. We\ngive an adiabaticity condition for the spectator field distribution to relax to\nthe de-Sitter equilibrium, and find that the de-Sitter approximation is never a\nreliable estimate for the typical distribution at the end of inflation for a\nquadratic spectator during monomial inflation. The existence of an adiabatic\nregime at early times can erase the dependence on initial conditions of the\nfinal distribution of field values. In these cases, spectator fields acquire\nsub-Planckian expectation values. Otherwise spectator fields may acquire much\nlarger field displacements than suggested by the de-Sitter equilibrium\nsolution. We quantify the information about initial conditions that can be\nobtained from the final field distribution. Our results may have important\nconsequences for the viability of spectator models for the origin of structure,\nsuch as the simplest curvaton models.\n