No Page Curves for the de Sitter Horizon

TL;DR

This work analyzes the fine-grained entropy of the de Sitter horizon by performing a partial dimensional reduction of dS$_3$ to JT gravity on dS$_2$ and coupling it to a thermal bath, enabling a tractable, time-dependent entropy calculation for an out-of-equilibrium Unruh-de Sitter state. The authors identify two decoupled regions—the I$^+$ boundary and a weakly gravitating region near the past cosmological horizon inside the static patch—where entropy can be computed using 2D CFT techniques and the island rule is invoked for the static patch. They find that the meta-observer at I$^+$ observes a unitary evolution without islands, while the static patch observer relies on an island to reproduce a Page curve; however, backreaction eventually forms a trapped region at the Page time, obstructing information recovery. These results illuminate observer-dependent unitarity in evaporating cosmological horizons, connect higher-dimensional baths to island calculations, and offer potential implications for inflationary cosmology and horizon holography in de Sitter space.

Abstract

We investigate the fine-grained entropy of the de Sitter cosmological horizon. Starting from three-dimensional pure de Sitter space, we consider a partial reduction approach, which supplies an auxiliary system acting as a heat bath both at future infinity and inside the static patch. This allows us to study the time-dependent entropy of radiation collected for both observers in the out-of-equilibrium Unruh-de Sitter state, analogous to black hole evaporation for a cosmological horizon. Central to our analysis in the static patch is the identification of a weakly gravitating region close to the past cosmological horizon; this is suggestive of a relation between observables at future infinity and inside the static patch. We find that in principle, while the meta-observer at future infinity naturally observes a pure state, the static patch observer requires the use of the island formula to reproduce a unitary Page curve. However, in practice, catastrophic backreaction occurs at the Page time, and neither observer will see unitary evaporation.

Figures (8)

• Figure 1: Global de Sitter is the surface of the hyperboloid (a). Time flows upwards; one angular coordinate is suppressed, such that each time-slice is actually a two-sphere (b). We split the spacetime into two, reducing over the red part to get JT gravity on dS$_2$. The green part is the remainder of dS$_3$, which takes on the role of the bath.
• Figure 2: The Penrose diagram of three- and two-dimensional pure de Sitter. For dS$_3$ each point represents a circle. The static patch for an observer at the south pole is indicated in orange; the dashed lines are the horizons. The Milne (future) patch is indicated in blue. For dS$_2$ we will make use of the fluctuating boundary geometry at $\mathcal{I}^{+}$ described by a Schwarzian action. $\mathcal{I}^{-}$ does not play a role in our considerations as we consider a quantum state that is singular at the past horizon.
• Figure 3: Penrose diagram of the Unruh state. The black bars denote zero one-point functions, whereas the arrows denote non-zero one-point functions of the stress tensor. Globally (red radiation), there are only left-moving modes and no right-moving modes. In the static patch (orange radiation), this corresponds to no outgoing radiation. This gets transferred to the Milne patch (blue radiation). As also elaborated upon in the main text, in global or Kruskal coordinates there is a flux of negative energy.
• Figure 4: A constant $\chi$ slice of the (renormalised) cylinder at $\mathcal{I}^{+}$ of three-dimensional de Sitter in our partial reduction approach. In the two-dimensional picture the radiation ends up at $\mathcal{I}^{+}$ (red), before evaporating into the bath (green), which is located along a higher dimension ($\varphi$). The transfer of radiation from dynamical gravity to bath corresponds to \ref{['eq:anglesolution']}. Note that as $u$ increases, $\alpha (u)$ will decrease such that we are indeed modelling an evaporating system. We included the higher-dimensional de Sitter region in opaque red to clarify the entanglement structure.
• Figure 5: The radiation entropy collected at $\mathcal{I}^{+}$. For this plot, we took the cut-off $\epsilon = 2 \ell e^{-2 G S_\text{dS}}$, and $\ell = G = 1$, which fixes $c = \frac{3}{2}$. The qualitative behaviour is the same for any $c$; this just determines the range of $u$.