Towards a gauge invariant volume-weighted probability measure for eternal inflation

TL;DR

Eternal inflation yields infinities and gauge-dependent cut-offs that blur probabilistic predictions across the multiverse. The paper proposes a stationary, gauge-invariant volume-weighted measure that ties probabilities to volume growth during inflation and the onset of stationarity, rendering results largely independent of time parametrization for $β>0$. In a simple three-minima model, the relative volumes obey $P(φ_1)/P(φ_5) = Γ_{32} e^{3N_{21}} / (Γ_{34} e^{3N_{45}})$, with the youngness paradox removed when clocks are reset at stationarity; the same principle extends to slow-roll stages and generalizes to temperatures via volume factors. This stationary approach offers a robust framework for comparing vacua in the landscape and clarifies when different volume-weighted measures agree, while pointing to extensions to more complex tunneling scenarios and slow-roll dynamics.

Abstract

An improved volume-weighted probability measure for eternal inflation is proposed. For the models studied in this paper it leads to simple and intuitively expected gauge-invariant results.

Figures (3)

• Figure 1: A potential with three dS minima and two sinks.
• Figure 2: Tunneling from the false vacuum to two different slow roll inflation regimes.
• Figure 3: Large green bubbles correspond to the bubbles containing the field $\phi_1$. Small red bubbles contain field $\phi_5$. The chain reaction of the bubble production starts at the time $t_{12}$ for the 'tree' of the bubbles of the field $\phi_1$, and at the time $t_{45}$ for the 'tree' of the bubbles of the field $\phi_5$. After this process starts, the number of the bubbles belonging to each 'tree' grows at the same rate, as $e^{3H_2\tau}$, where $\tau$ is the time starting from the moment when the first bubble of each type appears. Instead of comparing the number of the bubbles at the same moment of time $t$LLM, which does not make much sense for $t < max\{t_{21}, t_{45}\}$, we propose to reset the time for each type of the bubbles to the moment $\tau = 0$, when their number starts growing at the same rate, and compare them at the same time $\tau$. In other words, one should compare apples to apples, instead of comparing apples to the trunks of the trees. The results of this comparison in our simple model do not depend on initial conditions, on the time $\tau$, or on the choice of time parametrization.