New Scale Factor Measure

TL;DR

The paper introduces the New Scale Factor Cutoff, a new measure for probabilities in eternally inflating universes that remains well-defined even in contracting regions, improving upon previous measures.

Contribution

It proposes a novel regulator that extends the scale factor measure to all spacetime regions, including contracting and gravitationally bound areas.

Findings

The new measure is well-defined in entire spacetime.

It avoids divergences despite including infinite disconnected regions.

It combines advantages of previous cutoff measures.

Abstract

The computation of probabilities in an eternally inflating universe requires a regulator or "measure". The scale factor time measure truncates the universe when a congruence of timelike geodesics has expanded by a fixed volume factor. This definition breaks down if the generating congruence is contracting---a serious limitation that excludes from consideration gravitationally bound regions such as our own. Here we propose a closely related regulator which is well-defined in the entire spacetime. The New Scale Factor Cutoff restricts to events with scale factor below a given value. Since the scale factor vanishes at caustics and crunches, this cutoff always includes an infinite number of disconnected future regions. We show that this does not lead to divergences. The resulting measure combines desirable features of the old scale factor cutoff and of the light-cone time cutoff, while eliminating some of the disadvantages of each.