Particle creation by de Sitter black holes revisited

The creation of a thermal distribution of particles by a black hole is independent of the detail of gravitational collapse, making the construction of the eternal horizons sufficient to address the problem in asymptotically flat spacetimes. For eternal de Sitter black holes, however, earlier studies have shown the existence of both thermal and nonthermal particle creation, originating from the nontrivial causal structure of these spacetimes. Keeping this in mind, we consider this problem in the context of a quasistationary gravitational collapse occurring in a ($3+1$)-dimensional eternal de Sitter spacetime, settling down to a Schwarzschild--or Kerr--de Sitter spacetime, and we consider a massless minimally coupled scalar field. There is a unique choice of physically meaningful ``in'' vacuum here, defined with respect to the positive-frequency cosmological Kruskal modes localized on the past cosmological horizon C^-, at the onset of the collapse. We define our ``out'' vacuum at a fixed radial coordinate ``close'' to the future cosmological horizon, C⁺, with respect to positive-frequency outgoing modes written in terms of the ordinary retarded null coordinate, u. We trace such modes back to C^- along past-directed null geodesics through the collapsing body. Some part of the wave will be reflected back without entering it due to the graybody effect. We show that these two kinds of traced-back modes yield the two-temperature spectra and fluxes subject to the aforementioned ``in'' vacuum. Since the coordinate u used in the ``out'' modes is not well defined on a horizon, an estimate on how ``close'' we might be to C⁺ is given by estimating the backreaction. We argue that no other reasonable choice of the ``out'' vacuum would give rise to any thermal spectra. Our conclusions remain valid for all non-Nariai-class black holes, irrespective of the relative sizes of the two horizons.