Quasinormal modes for the SdS black hole : an analytical approximation scheme

TL;DR

The paper studies scalar perturbations in Schwarzschild-de Sitter ($\mathrm{SdS}$) spacetimes and proposes an analytic approximation to the SdS potential in the regime $M \ll l$. It forms the potential from two matched Poschl-Teller wells to reproduce the SdS asymptotics and yields closed-form QNM frequencies in two scale-separated families, tied to the horizon surface gravities $\alpha_b$ and $\alpha_c$. This reproduces the qualitative two-time-scale decay seen in numerical SdS studies, with intermediate-time QNMs governed by the black-hole horizon and late-time QNMs by the cosmological horizon, while highlighting incompleteness and angular-modes dependence as directions for refinement. The work provides a tractable framework to understand SdS perturbations and suggests future refinements to capture angular dependence and potential power-law tails, potentially informing time-domain behavior and holographic interpretations.

Abstract

Quasinormal modes for scalar field perturbations of a Schwarzschild-de Sitter (SdS) black hole are investigated. An analytical approximation is proposed for the problem. The quasinormal modes are evaluated for this approximate model in the limit when black hole mass is much smaller than the radius of curvature of the spacetime. The model mirrors some striking features observed in numerical studies of time behaviour of scalar perturbations of the SdS black hole. In particular, it shows the presence of two sets of modes relevant at two different time scales, proportional to the surface gravities of the black hole and cosmological horizons respectively. These quasinormal modes are not complete - another feature observed in numerical studies. Refinements of this model to yield more accurate quantitative agreement with numerical studies are discussed. Further investigations of this model are outlined, which would provide a valuable insight into time behaviour of perturbations in the SdS spacetime.