STATIONARY SOLUTIONS IN BRANS-DICKE STOCHASTIC INFLATIONARY COSMOLOGY

TL;DR

This work analyzes stationary solutions in Brans-Dicke stochastic inflation by studying diffusion of the inflaton and dilaton fields. It shows that standard Brans-Dicke models with chaotic potentials yield runaway, non-stationary probability distributions, but a non-minimal curvature coupling and radiative corrections can introduce dynamical cutoffs that generate stationary distributions $P_p(\phi,\sigma,t)$. In particular, a massive curvature-coupled inflaton yields a finite self-reproduction region and a stationary peak near $\phi=0$ with potentially exponentially large end-of-inflation Planck mass, while self-coupling can destabilize stationarity. One-loop corrections can produce a large-$\sigma$ maximum, reinforcing the stationary regime and providing a natural boundary for diffusion; conversely, non-perturbative jumps are shown to be strongly suppressed in the stationary BD context, contrasting with some Einstein-gravity scenarios. These results illuminate how diffusion-based cosmological measures interact with gravity theories to shape the distribution of fundamental scales and have implications for Planck-mass hierarchies and quantum cosmology interpretability.

Abstract

In Brans-Dicke theory the Universe becomes divided after inflation into many exponentially large domains with different values of the effective gravitational constant. Such a process can be described by diffusion equations for the probability of finding a certain value of the inflaton and dilaton fields in a physical volume of the Universe. For a typical chaotic inflation potential, the solutions for the probability distribution never become stationary but grow forever towards larger values of the fields. We show here that a non-minimal conformal coupling of the inflaton to the curvature scalar, as well as radiative corrections to the effective potential, may provide a dynamical cutoff and generate stationary solutions. We also analyze the possibility of large nonperturbative jumps of the fluctuating inflaton scalar field, which was recently revealed in the context of the Einstein theory. We find that in the Brans--Dicke theory the amplitude of such jumps is strongly suppressed.