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 $r_c\gg r_e,$ the field’s time evolution can be separated into three epochs. At times $t\ll t_c,$ 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 $t\lesssim t_c,$ the power-law behavior gives way to a faster, exponential decay. In this intermediate epoch, the conditions at radii $r\gtrsim r_e$ and $r\lesssim r_c$ both play an important role. Finally, at times $t\gg t_c,$ 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 ${\xi }_c=3/16\mathrm{}$, the field decays exponentially, with a decay constant that increases with increasing $\xi$. If $\xi >{\xi }_c,$ 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 $\xi$. We establish these properties using a combination of numerical and analytical methods.