Spacetime Entanglement Entropy of de Sitter and Black Hole Horizons

TL;DR

This paper calculates the covariant entanglement entropy for scalar fields at de Sitter and Schwarzschild de Sitter horizons, showing it is finite, independent of mass, and proportional to the horizon area.

Contribution

It extends the calculation of horizon entanglement entropy to de Sitter and Schwarzschild de Sitter spacetimes using covariant methods, demonstrating area proportionality.

Findings

Entanglement entropy is finite and well-defined for the static patches.

Entropy is independent of the scalar field's effective mass.

Entropy scales with the horizon area after angular mode cut-off.

Abstract

We calculate Sorkin's manifestly covariant entanglement entropy $\mathcal{S}$ for a massive and massless minimally coupled free Gaussian scalar field for the de Sitter horizon and Schwarzschild de Sitter horizons respectively in $d > 2$. In de Sitter spacetime we restrict the Bunch-Davies vacuum in the conformal patch to the static patch to obtain a mixed state. The finiteness of the spatial $\mathcal{L}^2$ norm in the static patch implies that $\mathcal{S}$ is well defined for each mode. We find that $\mathcal{S}$ for this mixed state is independent of the effective mass of the scalar field, and matches that of Higuchi and Yamamoto, where, a spatial density matrix was used to calculate the horizon entanglement entropy. Using a cut-off in the angular modes we show that $\mathcal{S} \propto A_{c}$, where $A_c$ is the area of the de Sitter cosmological horizon. Our analysis can be carried over to the black hole and cosmological horizon in Schwarzschild de Sitter spacetime, which also has finite spatial $\mathcal{L}^2$ norm in the static regions. Although the explicit form of the modes is not known in this case, we use appropriate boundary conditions for a massless minimally coupled scalar field to find the mode-wise $\mathcal{S}_{b,c}$, where $b,c$ denote the black hole and de Sitter cosmological horizons, respectively. As in the de Sitter calculation we see that $\mathcal{S}_{b,c} \propto A_{b,c}$ after taking a cut-off in the angular modes.