Proliferation of de Sitter Space

Abstract

I show that de Sitter space disintegrates into an infinite number of copies of itself. This occurs iteratively through a quantum process involving two types of topology change. First a handle is created semiclassically, on which multiple black hole horizons form. Then the black holes evaporate and disappear, splitting the spatial hypersurfaces into large parts. Applied to cosmology, this process leads to the production of a large or infinite number of universes in most models of inflation and yields a new picture of global structure.

Figures (4)

• Figure 1: Evolution of spacelike hypersurfaces of de Sitter space during the creation and subsequent evaporation of a single black hole. The spontaneous creation of a handle changes the spatial topology from spherical ($S^3$) to toroidal ($S^1 \times S^2$) with constant two-sphere radius. (The double-headed arrow indicates that opposite ends of the middle picture should be identified, closing the $S^1$.) If the quantum fluctuations are dominated by the lowest Fourier mode on the $S^1$, there will be one minimum and one maximum two-sphere radius, corresponding to a black hole (b) and a cosmological horizon (c). This resembles a 'wobbly doughnut' with cross-sections of varying thickness. As the black hole evaporates, the thinnest cross-section decreases in size. Finally the black hole disappears, i.e. the doughnut is pinched at its thinnest place and reverts to the original spherical topology.
• Figure 2: Penrose diagram for the process depicted in Fig. \ref{['fig-dissoc-1']}. The shaded region is the black hole interior. In the region marked by the square brackets the spatial topology is $S^1 \times S^2$, and opposite ends should be identified. The middle picture in Fig. \ref{['fig-dissoc-1']} corresponds to the handle nucleation surface shown here. After the black hole evaporates, a single de Sitter universe remains.
• Figure 3: Evolution of spacelike hypersurfaces of de Sitter space during the creation of a handle yielding multiple black holes ($n=2$) and their subsequent evaporation. This should be compared to Fig. \ref{['fig-dissoc-1']}. If the quantum fluctuations on the $S^1 \times S^2$ handle are dominated by the second Fourier mode on the $S^1$, there will be two minima and two maxima of the two-sphere radius, seeding to two black hole interiors (b) and two asymptotically de Sitter regions (c). This resembles a 'wobbly doughnut' on which the thickness of the cross-sections oscillates twice. As the black holes evaporate, the minimal cross-sections decrease. When the black holes disappear, the doughnut is pinched at two places, yielding two disjoint spaces of spherical topology, the daughter universes.
• Figure 4: Penrose diagram for the process depicted in Fig. \ref{['fig-dissoc-n']}. The shaded regions are the two black hole interiors. In the region marked by the square brackets the spatial topology is $S^1 \times S^2$, and opposite ends should be identified. The middle picture in Fig. \ref{['fig-dissoc-n']} corresponds to the horizon freezing surface shown here. After the black holes evaporate, two separate de Sitter universes remain.