De Sitter Holography and Entanglement Entropy

TL;DR

This work proposes a concrete dS_{d+1}/dS_d holographic duality in which two coupled, cutoff CFT sectors constrained by residual $d$-dimensional gravity encode de Sitter entropy. Using a tractable $d=2$ setup, the authors identify a finite Hilbert space per sector near the Cardy regime and argue that inter-sector interactions drive the reduced density matrix to maximal mixing, yielding a Renyi spectrum independent of n. The holographic RT/HRT calculations reproduce the Gibbons-Hawking entropy, including its numerical coefficient, and interpret it as the entanglement entropy between the two CFT sectors, with a volume-law behavior for subregions and a vanishing mutual information at leading order. The results extend to higher dimensions via the Hawking-Page transition and offer a cosmological realization of entropy-entanglement relations in holography, with several avenues for future exploration including FRW cosmologies and connections to deformations like $T\bar T$.

Abstract

We propose a new example of entanglement knitting spacetime together, satisfying a series of checks of the corresponding von Neumann and Renyi entropies. The conjectured dual of de Sitter in d+1 dimensions involves two coupled CFT sectors constrained by residual d-dimensional gravity. In the d=2 case, the gravitational constraints and the CFT spectrum are relatively tractable. We identify a finite portion of each CFT Hilbert space relevant for de Sitter. Its maximum energy level coincides with the transition to the universal Cardy behavior for theories with a large central charge and a sparse light spectrum, derived by Hartman, Keller, and Stoica. Significant interactions between the two CFTs, derived previously for other reasons, suggest a maximally mixed state upon tracing out one of the two sectors; we derive this by determining the holographic Renyi entropies. The resulting entanglement entropy matches the Gibbons-Hawking formula for de Sitter entropy, including the numerical coefficient. Finally, we interpret the Gibbons-Hawking horizon entropy in terms of the Ryu-Takayanagi entropy, and explore the time evolution of the entanglement entropy.

Figures (4)

• Figure 1: On the left, we show an extremal surface (in black) bisecting a spatial slice (blue) of the eternal AdS black hole in $d=2$. It corresponds to the entanglement entropy obtained by tracing over one of the CFTs in the thermofield double state (\ref{['BHcase']}). On the right, we depict a spatial slice of $dS_3$ (\ref{['dSslicing']}) at the moment of time symmetry $\tau=0$, bisected by the extremal surface (in black) at the ultraviolet slice $w=\pi\ell_{dS}/2$. Its area corresponds to the entanglement entropy obtained by tracing over one of the two coupled CFTs in the dual of de Sitter. Given the maximally mixed density matrix derived from the Renyi entropies, we match the entanglement entropy, including its numerical coefficient. The entropy of the full circle is also the Gibbons-Hawking horizon entropy for an observer ${\cal O}$ momentarily at the deep IR end of one or the other warped throat, at $w=0$ or $\pi \ell_{dS}$ and $\tau=0$. That observer has no interaction with the second CFT, and must trace over it. See Fig. \ref{['Penrose']} below for the corresponding space-time diagram.
• Figure 2: The Penrose diagram of $dS_{d+1}$, with the UV slice $w=\pi \ell_{dS}/2$ of the geometry indicated by the dashed grey line. The solid grey lines indicate the two deep IR regions, $w=0$ and $w=\pi\ell_{dS}$. The RT surface $S^{d-1}$ is in the center of the diagram, indicated as a dark blue point. Its area determines the entanglement entropy between the two matter sectors. and tracing out the second sector gives a maximally mixed density matrix $\rho_1$ for the first. It is also the Gibbons-Hawking horizon entropy for an observer momentarily at the deep IR end of the second warped throat at $\tau=0$, indicated in light blue. That observer has no interaction with the first matter sector and must trace it out.
• Figure 3: HRT extremal surface for an interval of size $\phi$ at a general time $\tau_0$.
• Figure 4: The left panel contains the $dS_2$ Penrose diagram; the vertical sides are identified. The dark blue line at $\tau=0$ corresponds to the central dot (a circle of radius $\ell_{dS}$) in the $dS_3$ diagram from Fig. \ref{['Penrose']}, reproduced here on the right. This full circle of size $2\pi\ell_{dS}$ is the horizon for the observer ${\cal O}$ on the far right, causally connected to it through the $dS_3$ bulk. Within the global $dS_2$ geometry, the segments with extremal area $\pi\ell_{dS}$ between the end points are the largest for which the endpoints can communicate, as derived in the text. On a late-time slice, indicated in purple, we show a string of causal segments whose extremal area is $\pi\ell_{dS}$ and entanglement entropy is $\pi c/6$.