The Phase Transition to Eternal Inflation

Abstract

For slow-roll inflation we study the phase transition to the eternal regime. Starting from a finite inflationary volume, we consider the volume of the universe at reheating as order parameter. We show that there exists a critical value for the classical inflaton speed, ˙φ 2 /H 4 = 3/(2π 2), where the probability distribution for the reheating volume undergoes a sharp transition. In particular, for sub-critical inflaton speeds all distribution moments become infinite. We show that at the same transition point the system develops a nonvanishing probability of having a strictly infinite reheating volume, while retaining a finite probability for finite values. Our analysis represents the exact quantum treatment of the system at lowest order in the slow-roll parameters and H 2 /M 2 Pl.1 Introduction and setup 1.1 Heuristic thoughts Consider a standard slow-roll inflationary model. At the classical level, the inflaton field φ gently rolls down its potential, and as long as the slow-roll conditions are met the universe inflates. Eventually, once the inflaton kinetic energy becomes comparable to the potential energy, inflation ends