Two regimes of asymptotic fall-off of a massive scalar field in the Schwarzschild - de Sitter spacetime

The decay behavior of a massless scalar field in the Schwarzschild-de Sitter spacetime is well-known to follow an exponential law at asymptotically late times t -> infinity. In contrast, a massive scalar field in the asymptotically flat Schwarzschild background exhibits a decay with oscillatory (sinusoidal) tails enveloped by a power law. We demonstrate that the asymptotic decay of a massive scalar field in the Schwarzschild-de Sitter spacetime is exponential. Specifically, if mu M >> 1, where mu and M represent the mass of the field and the black hole, respectively, the exponential decay is also oscillatory. Conversely, in the regime of small mu M , the decay is purely exponential without oscillations. This distinction in decay regimes underscores the fact that, for asymptotically de Sitter spacetimes, a particular branch of quasinormal modes, instead of a "tail, " governs the decay at asymptotically late times. There are two branches of quasinormal modes for the Schwarzschild-de Sitter spacetime: the modes of an asymptotically flat black hole corrected by a nonzero A term, and the modes of an empty de Sitter spacetime corrected by the presence of a black hole. We show that the latter branch is responsible for the asymptotic decay. When mu M is small, the modes of pure de Sitter spacetime are purely imaginary (nonoscillatory), while at intermediate and large mu M they have both real and imaginary parts, what produces the two pictures of the asymptotic decay. In addition, we show that the asymptotic decay of charged and higher -dimensional black hole is also exponential.