Interaction of Hawking radiation with static sources in de Sitter and Schwarzschild–de Sitter spacetimes

Abstract

We study and look for similarities between the response rates ${R}^{\mathrm{dS}}{(a}_{0},\ensuremath{\Lambda})$ and ${R}^{\mathrm{SdS}}{(a}_{0},\ensuremath{\Lambda},M)$ of a static scalar source with constant proper acceleration ${a}_{0}$ interacting with a massless, conformally coupled Klein-Gordon field (i) in de Sitter spacetime, in the Euclidean vacuum, which describes a thermal flux of radiation emanating from the de Sitter cosmological horizon and (ii) in Schwarzschild--de Sitter spacetime, in the Gibbons-Hawking vacuum, which describes thermal fluxes of radiation emanating from both the hole and the cosmological horizons, respectively, where $\ensuremath{\Lambda}$ is the cosmological constant and M is the black hole mass. After performing the field quantization in each of the above spacetimes, we obtain the response rates at the tree level in terms of an infinite sum of zero-energy field modes possessing all possible angular momentum quantum numbers. In the case of de Sitter spacetime, this formula is worked out and a closed, analytical form is obtained. In the case of Schwarzschild--de Sitter spacetime such a closed formula could not be obtained, and a numerical analysis is performed. We conclude, in particular, that ${R}^{\mathrm{dS}}{(a}_{0},\ensuremath{\Lambda})$ and ${R}^{\mathrm{SdS}}{(a}_{0},\ensuremath{\Lambda},M)$ do not coincide in general, but tend to each other when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Lambda}}0$ or ${a}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}.$ Our results are also contrasted and shown to agree (in the proper limits) with related ones in the literature.