The Volume of the Universe after Inflation and de Sitter Entropy

TL;DR

This paper analyzes the probability distribution of the universe's volume after inflation, establishing bounds related to de Sitter entropy and exploring the transition between eternal and non-eternal inflation regimes.

Contribution

It introduces new techniques for studying eternal inflation and establishes a quantum bound on the volume produced by slow-roll inflation based on de Sitter entropy.

Findings

Probability distribution peaks around classical e-foldings in non-eternal regime

Transition to eternal inflation involves a rapid decrease in finite volume probability

Derived a quantum bound on inflationary volume related to de Sitter entropy

Abstract

We calculate the probability distribution for the volume of the Universe after slow-roll inflation both in the eternal and in the non-eternal regime. Far from the eternal regime the probability distribution for the number of e-foldings, defined as one third of the logarithm of the volume, is sharply peaked around the number of e-foldings of the classical inflaton trajectory. At the transition to the eternal regime this probability is still peaked (with the width of order one e-folding) around the average, which gets twice larger at the transition point. As one enters the eternal regime the probability for the volume to be finite rapidly becomes exponentially small. In addition to developing techniques to study eternal inflation, our results allow us to establish the quantum generalization of a recently proposed bound on the number of e-foldings in the non-eternal regime: the probability for slow-roll inflation to produce a finite volume larger than e^(S_dS/2), where S_dS is the de Sitter entropy at the end of the inflationary stage, is smaller than the uncertainty due to non-perturbative quantum gravity effects. The existence of such a bound provides a consistency check for the idea of de Sitter complementarity.