Entanglement in De Sitter Space

TL;DR

The paper generalizes entanglement entropies in de Sitter space by formulating monolayer and bilayer holographic proposals that replace AdS boundaries with static-patch horizons. Using the bit-thread formalism, it computes entanglement entropies for pure de Sitter, Schwarzschild–de Sitter, and split-horizon scenarios, showing that both proposals agree in many cases but can disagree when horizons are subdivided. The key finding is that horizon splitting yields nontrivial, sometimes non-extensive entanglement behavior, challenging conventional thermodynamic-limit expectations in holographic settings. The work suggests testing these predictions in finite, all-to-all systems such as SYK to clarify the role of the thermodynamic limit and deepen understanding of horizon entanglement in de Sitter space.

Abstract

This paper expands on two recent proposals, \cite{Susskind:2021dfc}\cite{Susskind:2021esx} and \cite{Shaghoulian:2021cef}, for generalizing the Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi formulas to de Sitter space. The proposals (called the monolayer and bilayer proposals) are similar; both replace the boundary of AdS by the boundaries of static-patches--in other words event horizons. After stating the rules for each, we apply them to a number of cases and show that they yield results expected on other grounds. The monolayer and bilayer proposals often give the same results, but in one particular situation they disagree. To definitively decide between them we need to understand more about the nature of the thermodynamic limit of holographic systems.

Figures (18)

• Figure 1: The left panel shows the Penrose diagram for de Sitter space with the pode and antipode static patches shown shaded. The right panel shows a time-symmetric spatial slice which forms a $(D-1)$-sphere. The horizons in both panels are colored purple. In the right panel the horizon is the bifurcate horizon which forms a $(D-2)$-sphere.
• Figure 2: The generalized horizon of the pode patch contains the cosmic horizon as the largest component, and also horizons of black holes that reside in the patch.
• Figure 3: A space-like slice through de Sitter space. In the static patches the slice follows constant time ($t$) surfaces until it gets near the horizons and then bends to cross the horizons. The static patches appear as hemispheres of radius $R.$ The exterior region between the horizons bulges to a larger size due to the growth of de Sitter space in the exterior as one moves away from the horizons.
• Figure 4: Bit-threads emitted from a horizon. The maximum area-density bit-threads is $1/4G.$ Bit-threads may not cross a horizon.
• Figure 5: The Ryu-Takayanagi minimal area surface defines a bottleneck for bit-threads.