Global-Local Duality in Eternal Inflation

TL;DR

The paper resolves a key aspect of the measure problem in eternal inflation by proving that the global light-cone time cut-off and the local causal patch measure yield identical relative probabilities in the late-time attractor regime dominated by the longest-lived metastable vacuum. This equivalence is established through a direct geometric relation: an event lies in a causal patch if and only if its generating geodesic enters its causal future, linking ε(Q) on the initial surface to the patch-weight via t(Q) = -1/3 log ε(Q). The authors derive the necessary late-time exponential growth assumptions, extend the analysis to general landscapes of metastable vacua, and show that the equivalence persists across simple and complex vacua structures, subject to certain regularity conditions (notably excluding Λ = 0 interiors). The work reinforces holographic motivations for these cut-offs and provides a framework for interpreting measures in a broadly applicable, gravity-consistent manner, with implications for phenomenology and potential constraints on fundamental landscape dynamics.

Abstract

We prove that the light-cone time cut-off on the multiverse defines the same probabilities as a causal patch with initial conditions in the longest-lived metastable vacuum. This establishes the complete equivalence of two measures of eternal inflation which naively appear very different (though both are motivated by holography). The duality can be traced to an underlying geometric relation which we identify.

Figures (3)

• Figure 1: Geodesics (thin vertical lines) emanating from an initial surface $\Sigma_0$ define an ensemble of causal patches (the leftmost is shaded grey/light) with a particular mix of initial conditions. The causal patch measure assigns to the event $Q$ a weight proportional to the number of patches that contain $Q$. Notice that $Q$ is contained precisely in those causal patches whose generating geodesics (blue) enter the causal future of $Q$, $I^+(Q)$ (shaded green/dark). In the continuum limit, the weight of $Q$ is therefore proportional to the volume, $\epsilon(Q)$, of the projection of $I^+(Q)$ onto $\Sigma_0$. This observation is crucial to our proof of equivalence to the light-cone time cut-off. The light-cone time of $Q$ is defined as $t(Q)\equiv -\frac{1}{3} \log \epsilon(Q)$.
• Figure 2: A geodesic $g$ starting from an initial surface $\Sigma_0$ defines a causal patch $C$ (shaded region), event horizon, $E$, and initial patch $\sigma$.
• Figure 3: Top: An ensemble of causal patches (shaded triangles) can be represented in a single large geometry. Suppose that initial conditions require starting in one of two particular de Sitter vacua, with probability $p^{(0)}_1=0.25$ and $p^{(0)}_2=0.75$. Let $\Sigma_0$ be a spacelike hypersurface containing a very large number of both types of de Sitter horizon regions, so that we can choose large numbers $Zp^{(0)}_1$ (dashed) and $Zp^{(0)}_2$ (solid) of nonoverlapping initial patches. Then relative probabilities for events of type $I$ and $J$ are given directly by the ratio $N_I/N_J$ of the numbers of such events in the causal patch regions.---Conversely, any $\Sigma_0$ and set of geodesics emanating from it defines an ensemble of causal diamonds. Increasing the density of geodesics enlarges the ensemble (bottom); an event occuring, say, in two different patches counts twice. If each vacuum region contains many horizon volumes, this will not change the statistical properties of the ensemble.