Eternal observers and bubble abundances in the landscape

TL;DR

This work addresses how to define bubble abundances $p_j$ in eternally inflating landscapes with fully recyclable vacua. It demonstrates that the natural eternal-observer definition, $p_j \propto \lim_{\tau\to\infty} N_j(\tau)$, coincides with the GSPVW bubble-abundance prescription and the Easther–Lim–Martin formulation in the recyclable regime. By deriving the stationary occupancy $f_j \propto H_j^4 e^{S_j}$ and showing $p_j \propto \sum_i \lambda_{ji}$, it clarifies that abundances reflect transition weights rather than entropy-driven time fractions. The paper emphasizes the conceptual separation between time spent in a vacuum and the frequency of encountering vacua, reinforcing the use of $p_j$ for landscape-probability calculations in recyclable models.

Abstract

We study a class of ``landscape'' models in which all vacua have positive energy density, so that inflation never ends and bubbles of different vacua are endlessly ``recycled''. In such models, each geodesic observer passes through an infinite sequence of bubbles, visiting all possible kinds of vacua. The bubble abundance $p_j$ can then be defined as the frequency at which bubbles of type $j$ are visited along the worldline of an observer. We compare this definition with the recently proposed general prescription for $p_j$ and show that they give identical results.