We solve analytically the low-frequency s-wave dynamics of a massless scalar field propagating on a Schwarzschild-de Sitter black hole background. A rigorous application of the method of matched asymptotic expansions allows us to connect the scalar’s evolution in the proximity of the black hole horizon with that on cosmological scales. The scattering coefficients, greybody factors, and Wigner time delay are computed explicitly. We consider both small and large black holes, with black hole to cosmological horizon radii parametrically small and of order unity, respectively. This extends previous studies confined to the small black hole regime only. In addition, for small black holes, we perform a calculation that remains agnostic about the relative size between the ratio of the geometry’s horizons and the scalar’s frequency in units of the black hole radius. In this case, we find that they are interchangeable in the greybody factor, which is symmetric under $\omega \leftrightarrow 1/r_c$ (where $\omega$ is the scalar’s frequency and $r_c$ the cosmological horizon radius).

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• Classical black holes
• General relativity