Radiative falloff in Schwarzschild-de Sitter spacetime

We consider the evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. The field is non-minimally coupled to curvature through a coupling constant xi. The spacetime has two distinct time scales, t(e) = r(e)/c and t(c) = r(c)/c, where r(e) is the radius of the black-hole horizon, r(c) the radius of the cosmological horizon, and c the speed of light. When r(c) much greater than r(e). the field's time evolution can be separated into three epochs. At times t much less than t(c), the field behaves as if it were in pure Schwarzschild spacetime; the structure of spacetime far from the black hole has no influence on the evolution. In this early epoch, the field's initial outburst is followed by quasi-normal oscillations, and then by an inverse power-law decay. At times t less than or similar to t(c), the Fewer-law behavior gives way to a faster, exponential decay. In this intermediate epoch, the conditions at radii r greater than or similar to r(e), and r less than or similar to r(c), both play an important role. Finally, at times t much greater than t(c), the field behaves as if it were in pure de Sitter spacetime; the structure of spacetime near the black hole no longer influences the evolution in a significant way. In this late epoch, the field's behavior depends on the value of the curvature-coupling constant xi. If xi is less than a critical value xi(c)= 3/16, the held decays exponentially, with a decay constant that increases with increasing xi. If xi > xi(c), the field oscillates with a frequency Chat increases with increasing xi the amplitude of the field still decays exponentially, but the decay constant is independent of xi. We establish these properties using a combination of numerical and analytical methods. [S0556-2821(99)06416-4].