Measure Problem for Eternal and Non-Eternal Inflation

TL;DR

This paper examines how different probability measures in eternal inflation models respond to a parameter that interpolates between eternal and non-eternal inflation, revealing that only the stationary measure changes continuously.

Contribution

It introduces a toy model to compare four measures' predictions across eternal and non-eternal inflation, highlighting the unique continuity of the stationary measure.

Findings

Only the stationary measure's predictions change continuously with the parameter.

Proper-time and scale factor cutoff measures show discontinuous predictions.

Causal diamond measure's predictions are continuous only with sufficiently long slow-roll inflation.

Abstract

We study various probability measures for eternal inflation by applying their regularization prescriptions to models where inflation is not eternal. For simplicity we work with a toy model describing inflation that can interpolate between eternal and non-eternal inflation by continuous variation of a parameter. We investigate whether the predictions of four different measures (proper time, scale factor cutoff, stationary and causal {diamond}) change continuously with the change of this parameter. We will show that {only} for the stationary measure the predictions change continuously. For the proper-time and the scale factor cutoff, the predictions are strongly discontinuous. For the causal diamond measure, the predictions are continuous only if the stage of the slow-roll inflation is sufficiently long.