The stochastic spectator

Abstract

Abstract. 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.

Scalar correlation functions for a double-well potential in de Sitter space

“…For other techniques, see Refs. [17][18][19][20][21][22][23][24][25][26][27][28][29] The stochastic approach has become increasingly popular in recent years, likely due to its great efficacy, and in this vein we note the recent works [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Often the focus is on the local probability distribution of the field or on local expectation values, even though the correlation of fluctutations over space is arguably the more relevant object physically.…”

Scalar correlation functions in de Sitter space from the stochastic spectral expansion

“…The one-point probability distribution is well-known and widely used, see for example Ref. [45]. Investigating correlators via the stochastic formalism has been less common, however see [7,9,33,[46][47][48][49][50][51][52][53][54][55][56][57], even though it is the correlators that are more often directly related to observations.…”

Renormalisation group improvement in the stochastic formalism

“…Beyond this limit it is necessary to use (2.18) or (3.5) to maintain the correct RG scaling. As shown in reference [35], the true equilibrium may significantly deviate from the stationary solution whenever the spacetime is not exactly de Sitter. In situations where quantum corrections and running couplings are significant, it is therefore important to use the full equations (2.18) and (3.5) with the correct behaviour under the renormalisation group instead of the naive replacement.…”

“…In specific examples it has been shown to correctly reproduce the leading-log order IR results derived by other means [23][24][25][26][27][28][29]. Recent works addressing foundations of the stochastic approach include references [30][31][32][33][34][35][36][37][38]. During inflation light energetically subdominant scalars probe field values from the vacuum parametrically up to the Hubble scale H. Quantum corrections may induce significant running of couplings over this window and must be accounted in studying the dynamics.…”