Asymptotics of Killing horizons: A Characterization in de Sitter Spacetime

We prove that (n + 1)-dimensional de Sitter spacetime is locally the unique $(\mathrm{\Lambda }>0)$-vacuum spacetime whose Killing horizons intersect the conformal boundary $\mathcal{I}$. The intersection occurs at the isolated essential zeros of the extension of the Killing vector field to $\mathcal{I}$. We then reconstruct all de Sitter Killing vector fields from their boundary value and obtain a characterization of those admitting a Killing horizon in terms of a conformal invariant $\sigma$: $\sigma <0$ characterizes non-degenerate bifurcate horizons, whereas $\sigma =0$ yields degenerate non-bifurcate horizons. Finally, we derive an intrinsic, conformally invariant formula for the surface gravity of the horizons in de Sitter, ${\kappa }^2=\mathrm{\mid }\sigma \mathrm{\mid }$.