Limiting geometry and spectral instability in Schwarzschild--de Sitter spacetimes

TL;DR

This paper develops a unified geometric and numerical framework to study quasinormal modes in Schwarzschild--de Sitter spacetimes, revealing spectral instabilities and mode accumulation phenomena relevant for gravitational wave analysis.

Contribution

It introduces a hyperboloidal geometric approach and analytical mesh refinement for QNM calculations, enabling analysis of limiting behaviors and spectral stability in Schwarzschild--de Sitter spacetimes.

Findings

Recovered known QNM families in limiting regimes

Identified spectral instability effects at the de Sitter limit

Proposed a heuristic measure for QNM density and mode accumulation

Abstract

We revisit the quasinormal mode (QNM) problem in Schwarzschild--de Sitter spacetimes providing a unified infrastructure tailored for studying limiting configurations. Geometrically, we employ the hyperboloidal framework to explicitly implement Geroch's rigorous limiting procedures for families of spacetimes. This enables a controlled transition between Schwarzschild, de Sitter, and Nariai geometries. Numerically, we introduce the analytical mesh refinement technique into quasinormal mode calculations, successfully recovering -- within the appropriate limiting scenarios -- both known families of quasinormal modes: complex light ring modes and purely imaginary de Sitter modes. We interpret these results in terms of spectral instability, where the notions of stable and unstable modes depends on the specific spacetime limit under consideration. In the Schwarzschild limit, de Sitter modes appear as a destabilizing effect on the continuous branch cut at $\omega = 0$. Conversely, the branch cut can be understood as emerging from an infinite accumulation of discrete modes at $\omega = 0$ in the transitional regime. We propose a heuristic measure of QNM density to characterize this accumulation and highlight the need for a more rigorous study of potential branch cut instabilities -- especially relevant in the context of late-time gravitational wave signals. The proposed infrastructure provides a general and extensible framework for investigations in more complex spacetimes, such as Reissner--Nordstr\"om--de Sitter or Kerr--Newman--de Sitter.