Stochastic inflation revisited: non-slow-roll statistics and DBI inflation

Abstract

Stochastic inflation describes the global structure of the inflationary universe bymodeling the super-Hubble dynamics as a system of matter fields coupled togravity where the sub-Hubble field fluctuations induce a stochastic force into theequations of motion. The super-Hubble dynamics are ultralocal, allowing us toneglect spatial derivatives and treat each Hubble patch as a separate universe. Thisprovides a natural framework in which to discuss probabilities on the space ofsolutions and initial conditions. In this paper we derive an evolution equationfor this probability for an arbitrary class of matter systems, including DBI andk-inflationary models, and discover equilibrium solutions that satisfy detailedbalance. Our results are more general than those derived assuming slow roll or aquasi-de Sitter geometry, and so are directly applicable to models that do notsatisfy the usual slow-roll conditions. We discuss in general terms the conditionsfor eternal inflation to set in, and we give explicit numerical solutions of highlystochastic, quasi-stationary trajectories in the relativistic DBI regime. Finally, weshow that the probability for stochastic/thermal tunneling can be significantlyenhanced relative to the Hawking–Moss instanton result due to relativistic DBIeffects.