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. The spacetime has two distinct time scales, t₄=r₄/c and t₂=r₂/c, where r₄ is the radius of the black-hole horizon, r₂ the radius of the cosmological horizon, and c the speed of light. When r₂r₄, the field's time evolution can be separated into three epochs. At times tt₂, 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 tt₂, the power-law behavior gives way to a faster, exponential decay. In this intermediate epoch, the conditions at radii rr₄ and rr₂ both play an important role. Finally, at times tt₂, 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. If is less than a critical value ₂=3/16, the field decays exponentially, with a decay constant that increases with increasing. If >₂, the field oscillates with a frequency that increases with increasing ; the amplitude of the field still decays exponentially, but the decay constant is independent of. We establish these properties using a combination of numerical and analytical methods.