Islands in de Sitter space

TL;DR

The paper analyzes entanglement between a gravitating de Sitter universe and a disjoint non-gravitating system within 2d JT gravity, showing that replica wormholes induce an island contribution that competes with conventional CFT entropy to saturate entanglement growth (monogamy). By solving semiclassical dS2 JT gravity with backreaction, constructing the Penrose diagrams, and applying the island formula, it demonstrates Page-like entropy behavior in both decompactified and compact geometries, and identifies several island types whose dominance depends on temperature and topology. In the de Sitter context, the island construction reveals a finite de Sitter Hilbert space and clarifies how cosmological and black hole horizons jointly generate extremal islands, including a proposed interpretation for compact universes (a maximal interval minus a puncture) to recover sensible entropy limits. The results extend the island program to cosmological spacetimes, illuminate the role of auxiliary entanglement in closed universes, and connect to broader questions about information flow and unitarity in de Sitter quantum gravity.

Abstract

We consider black holes in 2d de Sitter JT gravity coupled to a CFT, and entangled with matter in a disjoint non-gravitating universe. Tracing out the entangling matter leaves the CFT in a density matrix whose stress tensor backreacts on the de Sitter geometry, lengthening the wormhole behind the black hole horizon. Naively, the entropy of the entangling matter increases without bound as the strength of the entanglement increases, but the monogamy property predicts that this growth must level off. We compute the entropy via the replica trick, including wormholes between the replica copies of the de Sitter geometry, and find a competition between conventional field theory entanglement entropy and the surface area of extremal "islands" in the de Sitter geometry. The black hole and cosmological horizons both play a role in generating such islands in the back-reacted geometry, and have the effect of stabilizing the entropy growth as required by monogamy. We first show this in a scenario in which the de Sitter spatial section has been decompactified to an interval. Then we consider the compact geometry, and argue for a novel interpretation of the island formula in the context of closed universes that recovers the Page curve. Finally, we comment on the application of our construction to the cosmological horizon in empty de Sitter space.

Figures (7)

• Figure 1: The Penrose diagram of the black hole with the dilaton profile \ref{['eq:generalsourceless']} ($\zeta=0$). The blue dot is the event horizon of the black hole at $\theta=\pi$, and the orange dot is the cosmological horizon at $\theta=0$. In this work, we will use a somewhat nonstandard convention where $\theta$ increases from right to left.
• Figure 2: The Penrose diagram of the backreacted black hole. As we increase the CFT temperature, the black hole interior region gets longer. The blue dot represents the apparent horizon of the black hole, which differs from the event horizons. The orange dots are the cosmological apparent horizons, which are shown as overlapping the cosmological event horizons, but in general there is a slight difference between their positions. In the high temperature limit, they coincide at the diagram boundary.
• Figure 3: The universal covering space of the Penrose diagram of the backreacted black hole. Black dots indicate that we can continue this pattern indefinitely.
• Figure 4: Two types of islands $C$ in the geometry without backreaction \ref{['eq:generalsourceless']}, with $\zeta=0$. Instead of the islands themselves, we draw complementary regions of these islands $\bar{C}$ on the Cauchy slice $\tau=0$ (green lines). Left: The complement $\bar{C}$ of a type I island only contains the cosmological horizon. Right: The complement $\bar{C}$ of a type II island approximately connects the black hole and cosmological horizons.
• Figure 5: Left: The causal diamond $D(\Sigma)$ in the spacetime without backreaction. Right: The same causal diamond $D(\Sigma)$ in the backreacted black hole in the high temperature limit. The causal diamond shrinks to an almost null line connecting the black hole event horizon and the cosmological event horizon.