Boundary definition of a multiverse measure

TL;DR

The paper addresses infinities in eternal inflation by introducing a boundary-centric light-cone time, defined via the boundary volume of future light-cones in a conformal frame with $R = \text{constant}$. It uses the Yamabe problem to fix the conformal factor, showing that in a tractable FRW bubble model the boundary becomes a unit round $S^3$ and yields a finite, holographically motivated measure. A boundary rate equation and attractor solution are derived, leading to a simple event-counting expression that reproduces the old light-cone time and causal patch measures in the homogeneous limit. The work further discusses extensions to inhomogeneous and singular boundary cases, arguing that the boundary-based construction provides a robust, UV-finite framework with phenomenological viability for predicting vacua distributions in the multiverse.

Abstract

We propose to regulate the infinities of eternal inflation by relating a late time cut-off in the bulk to a short distance cut-off on the future boundary. The light-cone time of an event is defined in terms of the volume of its future light-cone on the boundary. We seek an intrinsic definition of boundary volumes that makes no reference to bulk structures. This requires taming the fractal geometry of the future boundary, and lifting the ambiguity of the conformal factor. We propose to work in the conformal frame in which the boundary Ricci scalar is constant. We explore this proposal in the FRW approximation for bubble universes. Remarkably, we find that the future boundary becomes a round three-sphere, with smooth metric on all scales. Our cut-off yields the same relative probabilities as a previous proposal that defined boundary volumes by projection into the bulk along timelike geodesics. Moreover, it is equivalent to an ensemble of causal patches defined without reference to bulk geodesics. It thus yields a holographically motivated and phenomenologically successful measure for eternal inflation.

Figures (4)

• Figure 1: Constant light-cone size on the boundary defines a hypersurface of constant "light-cone time" in the bulk. The green horizontal lines show two examples of such hypersurfaces. They constitute a preferred time foliation of the multiverse. In the multiverse, there are infinitely many events of both type 1 and type 2 (say, two different values of the cosmological constant measured by observers). Their relative probability is defined by computing the ratio of the number of occurrences of each event prior to the light-cone time $t$, in the limit as $t\to \infty$.
• Figure 2: Conformal diagram of de Sitter space with a single bubble nucleation. The parent de Sitter space is separated from the daughter universe by a domain wall, here approximated as a light-cone with zero initial radius (dashed line). There is a kink in the diagram where the domain wall meets future infinity. This diagram represents a portion of the Einstein static universe, but the Ricci scalar of the boundary metric is not constant.
• Figure 3: (a) The parent de Sitter space with the future of the nucleation event removed and (b) the bubble universe are shown as separate conformal diagrams which are each portions of the Einstein static universe. After an additional conformal transformation of the bubble universe (shaded triangle), the diagrams can be smoothly matched along the domain wall. The resulting diagram (c) has a round $S^3$ as future infinity but is no longer a portion of the Einstein static universe.
• Figure 4: Conformal diagram of de Sitter space containing a bubble universe with $\Lambda=0$