A prescription for probabilities in eternal inflation

TL;DR

The paper addresses the challenge of assigning gauge-invariant probabilities to different constants of nature in eternally inflating spacetimes with multiple discrete vacua. It generalizes the spherical-cutoff method through a marker-based prescription, yielding a practical, gauge-invariant rule for relative probabilities across thermalized region types. The main result provides a general relation $\frac{P_1}{P_2} = \frac{p_1 n_2}{p_2 n_1} \frac{\nu_1}{\nu_2}$, with concrete instantiations showing $\frac{P_1}{P_2} = \frac{\lambda_1}{\lambda_2} \left( \frac{Z_{*1}}{Z_{*2}} \right)^3$ in open inflation. The work clarifies connections to epsilon-prescriptions and FP formalisms, discusses gauge considerations and limitations, and outlines pathways for numerical and analytic extensions in predicting the relative abundances of bubble universes.

Abstract

Some of the parameters we call ``constants of Nature'' may in fact be variables related to the local values of some dynamical fields. During inflation, these variables are randomized by quantum fluctuations. In cases when the variable in question (call it $χ$) takes values in a continuous range, all thermalized regions in the universe are statistically equivalent, and a gauge invariant procedure for calculating the probability distribution for $χ$ is known. This is the so-called ``spherical cutoff method''. In order to find the probability distribution for $χ$ it suffices to consider a large spherical patch in a single thermalized region. Here, we generalize this method to the case when the range of $χ$ is discontinuous and there are several different types of thermalized region. We first formulate a set of requirements that any such generalization should satisfy, and then introduce a prescription that meets all the requirements. We finally apply this prescription to calculate the relative probability for different bubble universes in the open inflation scenario.

Figures (5)

• Figure 1: Symmetric double well inflaton potential
• Figure 2: A numerical simulation of the spacetime structure of an inflating universe VVW. The simulation corresponds to a double-well inflaton potential, with two degenerate minima where the inflaton takes the values $\pm \eta$. Inflating regions are white, while thermalized regions with inflaton values equal to $+\eta$ and $-\eta$ are shown with different shades of grey .
• Figure 3: Asymmetric double well potential which is symmetric near the maximum in the range $-\phi_s < \phi < \phi_s$.
• Figure 4: Asymmetric inflaton potential with a metastable vacuum at the top, which replaces the quantum diffusion region of Fig. 3.
• Figure 5: Spacetime geometry of the bubble in co-moving coordinates. Here, $r=|{\bf\rm x}|$, and various $\tau=const.$ surfaces of the open Robertson-Walker universe inside the bubble are shown, as well as one of the $\xi=const.$ curves.