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A measure of the multiverse

Alexander Vilenkin

TL;DR

This paper addresses the measure problem in the inflationary multiverse, where naive volume-based priors yield gauge artifacts and dependence on initial conditions. It advocates a pocket-based measure in which the prior is set by bubble abundance via $p_j = \lim_{\epsilon \to 0} \frac{N_j(> \epsilon)}{N(> \epsilon)}$ and the observable probability is $P_j = P_j^{(prior)} f_j$, with $f_j$ encoding the number of observers per unit comoving volume; special cases yield explicit results such as $p_j \propto \sum_i \Gamma_{ji} e^{S_i}$. The approach also presents equivalent formulations (ELM) and a terminal-vacua extension (Bousso) and generalizes to continuous fields $X$ through $\hat{P}_j(X)$. Overall, it provides a gauge- and initial-condition-insensitive framework to make statistical predictions across the string landscape, with extensions to collisions, diffusion, and observational tests guiding future refinement.

Abstract

I review recent progress in defining a probability measure in the inflationary multiverse. General requirements for a satisfactory measure are formulated and recent proposals for the measure are clarified and discussed.

A measure of the multiverse

TL;DR

This paper addresses the measure problem in the inflationary multiverse, where naive volume-based priors yield gauge artifacts and dependence on initial conditions. It advocates a pocket-based measure in which the prior is set by bubble abundance via and the observable probability is , with encoding the number of observers per unit comoving volume; special cases yield explicit results such as . The approach also presents equivalent formulations (ELM) and a terminal-vacua extension (Bousso) and generalizes to continuous fields through . Overall, it provides a gauge- and initial-condition-insensitive framework to make statistical predictions across the string landscape, with extensions to collisions, diffusion, and observational tests guiding future refinement.

Abstract

I review recent progress in defining a probability measure in the inflationary multiverse. General requirements for a satisfactory measure are formulated and recent proposals for the measure are clarified and discussed.

Paper Structure

This paper contains 11 sections, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem with global-time measure
  3. General requirements
  4. A pocket-based measure
  5. Bubble abundance $p_j$
  6. Eternal observers
  7. An equivalent proposal
  8. Bousso's proposal
  9. The selection factor $f_j$
  10. Continuous variables
  11. Discussion

Figures (1)

  • Figure 1: A schematic conformal diagram for a comoving region in an eternally inflating universe. Bubbles of different vacua are represented by different shades of gray. The upper boundary of the diagram $i_+$ is the future timelike infinity. A surface of constant global time $\Sigma$ cuts through the entire region and intersects many bubbles.