Pure de Sitter space and the island moving back in time

TL;DR

The paper investigates entanglement between the interior and exterior of a cosmological horizon in pure de Sitter space by applying the island formula within a 2D JT-like reduction of a 3D de Sitter model with semi-classical corrections. It derives a Page curve bounded by the Gibbons-Hawking entropy and shows the emergence of an island whose location and time evolution differ from black hole cases, thus preserving no-cloning via entanglement wedge reconstruction and introducing a Hayden-Paskill–type scrambling time. The results rely on a controlled 2D/3D setup with Polyakov terms and an anchor-curve construction to mimic radiation exchange, and they suggest that the qualitative island phenomenology extends to higher dimensions. These findings enhance parallels between cosmological horizons and black holes and offer a framework for exploring de Sitter holography and quantum information in cosmology.

Abstract

Observers in de Sitter space can only access the space up to their cosmological horizon. Assuming thermal equilibrium, we use the quantum Ryu-Takayanagi or island formula to compute the entanglement entropy between the states inside the cosmological horizon and states outside, as a function of time. We obtain a Page curve that is bound at a value corresponding to the Gibbons-Hawking entropy. At this transition an 'island' forms, which is in a significantly different location as compared to when considering black hole horizons and even moves back in time. These differences turn out to be essential for non-violation of the no-cloning theorem in combination with entanglement wedge reconstruction. This consideration furthermore introduces the need for a scrambling time, the entropy dependence of which turns out to coincide with what is expected for black holes. The model we employ has pure three-dimensional de Sitter space as a solution. We dimensionally reduce to two dimensions in order to take into account semi-classical effects. Nevertheless, we expect the aforementioned qualitative features of the island to persist in higher dimensions.

Figures (9)

• Figure 1: The goal of this paper is to study the entanglement entropy between the states inside the cosmological horizon and the states outside. Using the island formula we find the Page curve given in Figure \ref{['fig:pagecurve']}.
• Figure 2: Left: Causal diagram of pure de Sitter space beyond two dimensions (more details in Section \ref{['sec:model']}). The vertical lines are identified as the North and South pole. Right: Causal diagram of pure de Sitter space in two dimensions. Horizontal slices represent a circle. In both figures the diagonal lines indicate cosmological horizons and horizontal lines $\mathcal{I}^{\pm}$.
• Figure 3: In both figures the pink dot indicates the location of the extremal surface, the green dot indicates the location of the anchor point on the anchor curve (green line). The orange arrows denote movement directions in time and the brown wavy arrows represent semi-classical radiation. Left: A Penrose diagram of an eternal black hole in flat space is shown Gautason:2020tmk where the horizontal gray wavy line denote the location of the singularities and the diagonal dashed lines represent the horizon. Right: Penrose diagram of pure de Sitter beyond two dimensions. The vertical lines are the locations of the poles and can be reached in finite time, the diagonal lines represent the cosmological horizon and the horizontal lines represent $\mathcal{I}^{\pm}$. For more details see Section \ref{['sec:model']}.
• Figure 4: Conformal diagram for the three-dimensional de Sitter space, which is inherited by the two-dimensional de Sitter space. The transverse space at every point is a circle. The Kruskal coordinates in the north pole wedge are given in \ref{['eq:conf']}.
• Figure 5: Penrose diagram of semi-classical de Sitter space with radiation in the North pole wedge defined with respect to the Hartle-Hawking vacuum. The dilaton $\Phi$ is weakly coupled in all these regions.