Complementarity in the Multiverse

TL;DR

This paper analyzes the measure problem in eternal inflation by introducing a holographic UV-IR relation that defines a preferred global time via a boundary cut-off $\epsilon$, leading to the light-cone time measure. It demonstrates that the light-cone construction is a special case of Garriga-Vilenkin (GV) measures with a particular $\lambda(p)$, and that, in broad regions, this global measure is equivalent to the causal patch measure. Constant $\lambda$ reproduces the scale-factor cut-off, while allowing $\lambda$ to vary per event type yields simple mappings between GV, SF, and the light-cone approach, clarifying the relations among global and local descriptions. The results establish a multiverse complementarity between global holographic and local causal-patch pictures and point to open questions about hat/singular domains and a possible holographic boundary unification.

Abstract

In the multiverse, as in AdS, light-cones relate bulk points to boundary scales. This holographic UV-IR connection defines a preferred global time cut-off that regulates the divergences of eternal inflation. An entirely different cut-off, the causal patch, arises in the holographic description of black holes. Remarkably, I find evidence that these two regulators define the same probability measure in the multiverse. Initial conditions for the causal patch are controlled by the late-time attractor regime of the global description.

Figures (8)

• Figure 1: A global cut-off on the multiverse. The curvy lines represent hypersurfaces of constant $t$, with $t\to\infty$ at the top of the diagram. The relative probability of events 1 and 2 is defined by $p_1/p_2=\lim_{t\to\infty} N_1(t)/N_2(t)$, where $N_i(t)$ is the number of times a given event has occurred by the time $t$. This definition is ambiguous, because it depends on the choice of foliation, i.e., of the time coordinate $t$.
• Figure 2: The future light-cone of any event (black dots) defines a scale on the asymptotic boundary of the multiverse (top edge). (Like in AdS, this scale is physically infinite but can be regulated as described in the text.) Conversely, constant light-cone size on the boundary defines a hypersurface of constant "light-cone time" in the bulk (green horizontal lines). Taking the boundary scale to zero generates a preferred foliation of the multiverse. Remarkably, the resulting global probability measure is equivalent to the causal patch measure (below).
• Figure 3: The causal patch measure abandons the global description of the multiverse. Relative probabilities are defined by ratios of the expected numbers of events of different types taking place within the causal past of a single worldline (shaded region), averaging over possible histories and initial conditions. (The darker-shaded narrow triangle is discussed in Sec. 5.)
• Figure 4: Every point $p$ in Anti-de Sitter space is associated to a boundary scale $\delta$. This relation is completely determined by causality (left; only a small portion of the boundary is shown). It implies the well-known UV-IR connection of the AdS/CFT correspondence (right; here the global geometry is shown).
• Figure 5: The boundary scale $\epsilon$ associated with the multiverse event $p$ is defined as the volume (thick bars), on $\Sigma_0$, of those geodesics (thin vertical lines) that enter the future of $p$ (shaded). Other features in this diagram are discussed in Sec. 3.2.