Scalar field scattering in a Schwarzschild-de Sitter geometry

TL;DR

This work develops a rigorous analytic framework for low-frequency, radial $l=0$ scalar scattering on Schwarzschild-de Sitter backgrounds using matched asymptotic expansions. By treating small and large black-hole regimes separately, the authors derive explicit transmission and reflection coefficients, the greybody factor $\Gamma(\omega)$, and the Wigner time delay, revealing a symmetry under $\alpha \leftrightarrow \epsilon$ and showing that the zero-frequency greybody factor remains finite for small black holes. The analysis confirms that the leading low-frequency greybody factor matches across hole sizes, and provides detailed expressions for phase shifts and time delays arising from the de Sitter cosmological region. These results illuminate the interplay between near-horizon and cosmological dynamics in SdS spacetimes and offer a solid analytic baseline for future work including backreaction and the study of massive or non-minimally coupled fields.

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 $ω\leftrightarrow 1/r_c$ (where $ω$ is the scalar's frequency and $r_c$ the cosmological horizon radius).

Figures (2)

• Figure 1: The plots show the agreement between our 'near' and 'far' analytical solutions and a numerical one for $\phi(x)$ in the small black-hole regime. From left to right, we have $\alpha\ll\epsilon$ , $\alpha\sim\epsilon$ , $\epsilon\ll\alpha$ . Specifically, panels (a) and (d) correspond to $(\epsilon,\alpha)=(10^{-2},10^{-5})$ , panels (b) and (e) to $(\epsilon,\alpha)=(10^{-3},10^{-3})$ , (c) and (f) to $(\epsilon,\alpha)=(10^{-5},10^{-3})$ . The top and bottom rows, respectively, show the real and imaginary parts of $\phi$. The thick black curve represents the numerical solution, the dashed blue curve the near-region asymptotics, and the dotted orange curve corresponds to the far-region asymptotics.
• Figure 2: The plots show the agreement, in the large black-hole regime, between a numerical solution for $\phi(x)$ and our analytical approximations obtained with matched asymptotic expansions, for parameters $(\epsilon,\alpha)=(10^{-4},0.9)$ . The left and right panels, respectively, show the real and imaginary parts of $\phi$ as a function of $x$ . The thick black curve represents the numerical solution, the dashed blue curve the near-region approximation, the dot-dashed orange curve the static-region approximation, while the dashed red curve corresponds to the cosmological-region approximation.