Quantum Global Structure of de Sitter Space

TL;DR

This paper analyzes the global structure of de Sitter space in the semi-classical and one-loop quantum gravity regime. It shows that neutral black hole nucleation causes fragmentation into disconnected daughter universes, while magnetically charged black holes generate a necklace of de Sitter beads joined by near-extremal throats; with sufficiently large charge, an unbounded number of beads can form, leaving future infinity connected only through throats. The approach combines an analysis of Reissner–Nordström–de Sitter solutions, Charged Nariai geometry, a two-dimensional dilaton gravity model with one-loop back-reaction, linear perturbation theory, and Euclidean instanton methods to map the parameter space and late-time evolution. The results have implications for holographic descriptions of quantum gravity in de Sitter space and illuminate how global versus local perspectives shape the interpretation of cosmological topology changes.

Abstract

I study the global structure of de Sitter space in the semi-classical and one-loop approximations to quantum gravity. The creation and evaporation of neutral black holes causes the fragmentation of de Sitter space into disconnected daughter universes. If the black holes are stabilized by a charge, I find that the decay leads to a necklace of de Sitter universes (`beads') joined by near-extremal black hole throats. For sufficient charge, more and more beads keep forming on the necklace, so that an unbounded number of universes will be produced. In any case, future infinity will not be connected. This may have implications for a holographic description of quantum gravity in de Sitter space.

Figures (10)

• Figure 1: Penrose diagram of classical de Sitter space. The vertical lines are the origins of polar coordinates on opposite poles of the spatial three-spheres. The diagonal lines are the event horizons of observers on these poles.
• Figure 2: The three-step process by which de Sitter space proliferates into disconnected daughter universes. On a de Sitter background with $S^3$ spacelike sections, a neutral Nariai solution ($S^1 \times S^2$) nucleates (the arrows indicate that opposite ends should be identified, to form the $S^1$). By a classical instability, the degenerate Nariai geometry decays into a necklace of $n$ black holes and $n$ de Sitter regions ('beads'). Here $n=2$, and the spatial section was chosen so that it penetrates neither the black hole interiors nor the de Sitter regions; this means that the minimal two-spheres correspond to the black hole horizons (b), and the maximal two-spheres correspond to the cosmological horizons (c). When the black holes evaporate, the beads disconnect, and $n$ separate de Sitter universes remain. The corresponding Penrose diagram is shown in Fig. \ref{['fig-cp-prolneut']}.
• Figure 3: Penrose diagram for the proliferation process depicted in Fig. \ref{['fig-steps']}. Singularities are indicated by dashed lines. In the region marked by the square brackets the spatial topology is $S^1 \times S^2$, and opposite ends should be identified. Perturbations of the two-sphere radius first oscillate, then freeze out, seeding black hole interiors (light gray) and de Sitter regions (dark grey). After the black holes evaporate, two separate de Sitter universes remain.
• Figure 4: Charged Nariai black holes nucleate semiclassically in de Sitter space. Their spacelike sections are $S^1 \times S^2$. The one-sphere expands exponentially, and the two-sphere has a constant radius. The magnetic field loops around the one-sphere and prevents it from being pinched. Quantum fluctuations lead to the formation of $n$ black holes and $n$ de Sitter regions (here $n=2$). If the charge is small, the black holes evaporate until they are nearly extremal (see Fig. \ref{['fig-cp-prolsub']} for a Penrose diagram). For supercritical charge, the black holes will grow, approaching the radius of the Charged Nariai solution. The regions between a black hole and a cosmological horizon remain nearly degenerate. Small perturbations can produce more black hole interiors and de Sitter regions there. Iteratively, an infinite number of beads will form. The corresponding Penrose diagram is shown in Fig. \ref{['fig-cp-prolsup']}.
• Figure 5: Penrose diagram for the decay of de Sitter space into $n$ de Sitter regions (dark grey) separated by $n$ lukewarm black holes (light gray); here $n=2$. Singularities are indicated by dashed lines. The upper part of the diagram is a sequence of $n$ Reissner-Nordström-de Sitter diagrams (Fig. \ref{['fig-cp-rnds-tr']}). Future infinity fragments, but in contrast to the neutral case (Fig. \ref{['fig-cp-prolneut']}), space remains topologically connected. This diagram corresponds to the subcritical branch of Fig. \ref{['fig-structure']}.