Making Predictions in Eternally Inflating Universe

TL;DR

The paper tackles how to assign coordinate-independent probabilities to different low-energy constants realized in eternally inflating spacetimes, avoiding the strong $t$-dependence of equal-time volume comparisons. It introduces an epsilon-cutoff, observer-based regularization that yields a robust volume ratio between thermalized regions, with the dominant contribution governed by the slow-roll expansion factor $Z_*$ and modest sensitivity to the time parametrization. It also analyzes the spectrum of density fluctuations seen by a typical observer, showing a shifted Gaussian for $\delta\varphi$ with a small mean and rms given by the Hubble scale, and contrasts this with prior equal-time results. The approach has implications for predicting which constants are more probable across a multiverse and for comparing volumes across disconnected universes, though sharp predictions require significant asymmetry between competing minima.

Abstract

Eternally inflating universes can contain large thermalized regions with different values of the constants of Nature and with different density fluctuation spectra. To find the probability for a `typical' observer to detect a certain set of constants, or a certain fluctuation spectrum, one needs to compare the volumes occupied by different types of regions. If the volumes are taken on an equal-time hypersurface, the results of such a comparison are extremely sensitive to the choice of the time variable t. Here, I propose a method of comparing the volumes which is rather insensitive to the choice of t. The method is then applied to evaluate the relative probability of different minima of the inflaton potential and the probability distribution for the density fluctuation spectra.