Properties of the scale factor measure

TL;DR

The paper analyzes the scale factor measure in eternal inflation, clarifying its local reformulation in the no-collapse regime and deriving the rate-equation framework that governs vacuum volumes and observer counts. It demonstrates that, without collapse, the measure shares favorable properties with the causal diamond measure, including absence of Boltzmann brains and suppression of excessive volume weighting, and it yields a Lambda distribution broadly compatible with observations. However, incorporating collapsed regions drastically changes the Lambda prediction, pushing the peak to much larger values tied to structure-formation timescales, and highlighting potential tensions with the observed cosmological constant. The authors explore possible improvements to the collapse treatment and contrast Boltzmann-brain predictions with those of the causal diamond measure, finding that initial conditions and the handling of collapsed regions are crucial for the scale factor measure’s phenomenology and its viability as a measure of cosmological probabilities.

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.

Figures (2)

• Figure 1: Evolution of the scale factor cutoff during structure formation. The constant scale factor time surface $\Sigma_1$ lies in the early, approximately homogeneous universe and coincides with a surface of constant FRW time $\tau$. As density perturbations grow, some geodesics decouple from the Hubble flow, stop expanding, and become trapped in collapsed regions such as galaxies. If the scale factor cutoff exceeds the largest scale factor ever reached along such geodesics, then the rule of De Simone et al. requires that their entire future evolution be included. (A similar result obtains if the local formulation of the scale factor measure is used as a general definition; see Sec. 5.4.) Therefore, the later cutoff surface $\Sigma_2$ no longer agrees everywhere with a constant FRW time surface; it includes the entire future of collapsed regions (green/gray), which show up as spikes since the figure is drawn in comoving coordinates.
• Figure 2: The green (light shaded) slices are surfaces of constant scale factor time, $\Sigma_\eta$; they have fixed comoving size but increasing physical volume. Fat geodesics (purple, dark shaded) have fixed physical width and thus decreasing comoving size. If the initial slice is chosen in the attractor regime, the fat-geodesics define a representative finite sample of the total four-volume. Thus, the results of the scale factor measure can be reproduced by following a single geodesic of fixed width, starting in the longest-lived metastable vacuum.