Holographic probabilities in eternal inflation

TL;DR

This work addresses the ambiguity of probabilities in eternal inflation by adopting a local holographic viewpoint centered on a single causal diamond. It develops a matrix/tree formalism to compute prior probabilities of entering vacua, providing a general pruned-tree prescription that unifies terminal and cyclic landscapes. It then proposes an entropy-based weighting, w_i = ΔS(i), to estimate the likelihood of observer emergence without anthropic assumptions, yielding prior-free predictions and thermodynamic bounds. The approach clarifies how finite holographic bounds and thermodynamics constrain the landscape and observer-related probabilities, with implications for how we interpret the string landscape and cosmological measures.

Abstract

In the global description of eternal inflation, probabilities for vacua are notoriously ambiguous. The local point of view is preferred by holography and naturally picks out a simple probability measure. It is insensitive to large expansion factors or lifetimes, and so resolves a recently noted paradox. Any cosmological measure must be complemented with the probability for observers to emerge in a given vacuum. In lieu of anthropic criteria, I propose to estimate this by the entropy that can be produced in a local patch. This allows for prior-free predictions.

Figures (3)

• Figure 1: A landscape with two metastable vacua and one terminal vacuum. The tree on the left corresponds to a worldline starting in vacuum $A$ (the initial vacuum, or root). The tree on the right starts with vacuum $B$. The unnormalized probability for vacuum $i$ is obtained by computing the probability for each path leading up from the root to $i$ (the product of the numbers along the path), and summing over all paths.
• Figure 2: Probabilities are easier to compute from the pruned tree, shown left for the ABZ model, with initial vacuum $A$. One reads off readily that ${\cal P}_A = \epsilon$, ${\cal P}_B=1$, and ${\cal P}_Z=1-\epsilon$, which need only be normalized. Right: The full tree can be recovered by iterating the pruned tree. Each iteration changes all raw probabilities by the same factor, leaving the normalized probabilities invariant.
• Figure 3: A landscape without terminal vacua. For each initial vacuum, a pruned tree is shown. For example, summation over paths in the left tree yields ${\cal P}_A = 1$, ${\cal P}_B = 1/\epsilon$, ${\cal P}_C = (1-\epsilon)/\epsilon$. After normalization, all pruned trees yield the same probabilities.