The Price of Curiosity: Information Recovery in de Sitter Space

TL;DR

This work applies the island formula to Jackiw-Teitelboim gravity in de Sitter space to study whether a static observer can recover information from behind the cosmological horizon. In a non-equilibrium setup where radiation is collected inside the static patch, the authors analyze two 2D JT models (full and half reductions) and show that an island forms prior to the Page time in the full reduction, producing a Page-like entropy evolution, while no controlled island contribution appears in the half reduction. Backreaction from the collected radiation drives the cosmological horizon dynamics and, crucially, leads to the formation of a singularity outside the static patch, ensuring no violation of the no-cloning principle. The results connect de Sitter entropy to the horizon of a single observer and suggest a holographic/complementarity picture where gravitational physics in a static patch is dual to physics at the horizon, with a singularity acting as a safeguard against information duplication.

Abstract

Recent works have revealed that quantum extremal islands can contribute to the fine-grained entropy of black hole radiation reproducing the unitary Page curve. In this paper, we use these results to assess if an observer in de Sitter space can decode information hidden behind their cosmological horizon. By computing the fine-grained entropy of the Gibbons-Hawking radiation in a region where gravity is weak we find that this is possible, but the observer's curiosity comes at a price. At the same time the island appears, which happens much earlier than the Page time, a singularity forms which the observer will eventually hit. We arrive at this conclusion by studying Jackiw-Teitelboim gravity in de Sitter space. We emphasize the role of the observer collecting radiation, breaking the thermal equilibrium studied so far in the literature. By analytically solving for the backreacted geometry we show how an island appears in this out-of-equilibrium state.

Figures (10)

• Figure 1: Left: Constant time slice of de Sitter space in static coordinates. The polar angle $\theta$ runs from $-\pi/2$ to $+\pi/2$ and the azimuthal angle $\phi=\phi+ 2\pi$. Right: To obtain a two-dimensional dilaton gravity model we perform a reduction over the $S^1$ parametrized by $\phi$. We consider both a "half reduction" where we restrict $\theta$ to run from $-\pi/2$ to $+\pi/2$ and a "full reduction" where $\theta=\theta+2\pi$. The circumference of the orange $S^1$ indicates the size of the dilaton, which becomes negative in the red dotted region in the full reduction.
• Figure 2: Penrose diagram of two-dimensional de Sitter space obtained as a half reduction where we restrict $\Phi\geq0$ and impose reflecting boundary conditions at $\Phi=0$. The static coordinates \ref{['eq:staticmetric']} cover the two static patches shaded in blue and the coordinates $x^\pm$ (see \ref{['eq:KruskalMetric']}) cover both the blue and orange shaded regions.
• Figure 3: Penrose diagram of two-dimensional de Sitter space obtained as a full reduction in the near-horizon limit of a Nariai black hole in four dimensions. The static coordinates \ref{['eq:staticmetric']} cover the two static patches shaded in blue and the coordinates $x^\pm$ (see \ref{['eq:KruskalMetric']}) cover both the blue and orange shaded regions. The white regions are the black hole interior and the places where $\Phi\to-\infty$ are its singularities. The left and right vertical lines of the diagram are identified.
• Figure 4: Part of the Penrose diagram covered by $x^\pm$ coordinates in the non-equilibrium state \ref{['eq:UnruhStress']} with $t_+=0$. The red line indicates the stress tensor singularity at $x^+=0$. The region shaded gray is a trapped region.
• Figure 5: Using the formula for the fine-grained entropy, we compute the entropy $S(R)$ of a weakly-gravitating region $R=[A,A']$ which contains radiation emanating from the past horizon. Because the radiation is entangled with the region beyond the future horizon, we allow for a possible island $I$ that contributes to $S(R)$.