Geodesic measures of the landscape
Vitaly Vanchurin
TL;DR
The paper addresses the challenge of predicting bubble abundances in the landscape of eternal inflation with numerous vacua, including terminal states. It classifies geodesic-based measures into Single Geodesic (SG) and Ensemble of Geodesics (EG), and introduces three concrete measures: the S measure from a single eternal geodesic, the Eternal measure, and the Recursive measure, each with explicit Markov-process formulations. By deriving measure equations using transition matrices $\mathbf{T}$ and $\mathbf{R}$ and a normalization operator $\mathbf{N}[\cdot]$, the authors show how initial-condition sensitivity can be mitigated in the S measure and how different initial-choice prescriptions in EG lead to distinct bubble distributions $\mathbf{p}$. A toy three-vacuum example demonstrates how $\mathbf{p}_{s}$, $\mathbf{p}_{e}$, and $\mathbf{p}_{r}$ can diverge from the CH measure, emphasizing the role of observer weighting and initial-condition interpretation in predicting observable vacua. Overall, the work provides a unified, mathematically explicit framework for geodesic-based measures in eternal inflation and clarifies how initial conditions influence predictions across SG and EG schemes.
Abstract
We study the landscape models of eternal inflation with an arbitrary number of different vacua states, both recyclable and terminal. We calculate the abundances of bubbles following different geodesics. We show that the results obtained from generic time-like geodesics have undesirable dependence on initial conditions. In contrast, the predictions extracted from ``eternal'' geodesics, which never enter terminal vacua, do not suffer from this problem. We derive measure equations for ensembles of geodesics and discuss possible interpretations of initial conditions in eternal inflation.