  We consider the time evolution of a scalar field propagating in
Schwarzschild-de Sitter spacetime. At early times, 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 intermediate times, the power-law behavior gives
way to a faster, exponential decay. At late times, 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 3/16, the field decays
exponentially, with a decay constant that increases with increasing xi. If xi >
3/16, the field oscillates with a frequency that increases with increasing xi;
the amplitude of the field still decays exponentially, but the decay constant
is independent of xi.
