Eternal inflation and localization on the landscape

Abstract

We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential $V(φ)=V_{0}+δV(φ)$, where $|δV(φ)|\ll V_{0}$ is random. We find that the diffusion of the distribution function $ρ(φ,t)$ of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At $t \to \infty$ only the localized part of the distribution function $ρ(φ, t)$ survives which leads to dynamical selection principle on the landscape. The probability to measure any but a small value of the cosmological constant in a given Hubble patch on the landscape is exponentially suppressed at $t\to \infty$.