Properties of the scale factor measure

TL;DR

The paper reformulates the scale factor measure as a local measure in expanding regions, explaining its similarities to the causal diamond measure and analyzing its implications for the cosmological constant and primordial density contrast.

Contribution

It introduces a local formulation of the scale factor measure and compares its properties to the causal diamond measure, highlighting differences in probability assignments.

Findings

Both measures are free of Boltzmann brains.

The scale factor measure assigns a smaller probability to the observed cosmological constant.

Probability decreases with the inverse sixth power of the primordial density contrast.

Abstract

We show that in expanding regions, the scale factor measure can be reformulated as a local measure: Observations are weighted by integrating their physical density along a geodesic that starts in the longest-lived metastable vacuum. This explains why some of its properties are similar to those of the causal diamond measure. In particular, both measures are free of Boltzmann brains, subject to nearly the same conditions on vacuum stability. However, the scale factor measure assigns a much smaller probability to the observed value of the cosmological constant. The probability decreases further, like the inverse sixth power of the primordial density contrast, if the latter is allowed to vary.