Scalar field scattering in a Schwarzschild-de Sitter geometry

TL;DR

This paper analytically investigates the low-frequency scattering of a massless scalar field in Schwarzschild-de Sitter spacetime, providing explicit calculations of scattering coefficients and extending previous work to larger black holes.

Contribution

It introduces a rigorous matched asymptotic expansion method to analyze scalar scattering across different black hole sizes, including cases where horizon scales are comparable to scalar frequencies.

Findings

Explicit expressions for greybody factors and time delays.

Symmetry of greybody factors under frequency and horizon radius interchange.

Extension of previous small black hole studies to larger black holes.

Abstract

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. When the two are comparable, 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).