Spacetime entanglement entropy of de Sitter and black hole horizons

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 (2018 Phys. Rev. D 98 065014), 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 the boundary conditions of Qiu and Traschen (2020 Class. Quantum Grav. 37 135012) 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.