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

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 $ω= 0$. Conversely, the branch cut can be understood as emerging from an infinite accumulation of discrete modes at $ω= 0$ 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--Nordström--de Sitter or Kerr--Newman--de Sitter.

Figures (9)

• Figure 1: Penrose diagrams for the Schwarzschild--de Sitter spacetime and their respective limits achieved within the hyperboloidal framework as a natural implementation of Geroch’s geometric limiting procedure Geroch:1969ca. Solid points represent surfaces with a fixed coordinate value, whereas surfaces with an empty dot move freely in the grid. In all cases, $\mathscr{I}^+$ is fixed at a coordinate location $\sigma =0$. Top: the Schwarzschild scenario. The black hole horizon fixes the spacetime length scale $\lambda = r_H$. The event horizon ${\cal H}^+$ and the singularity are fixed, but the cosmological horizon is free. The corresponding limit $\eta \rightarrow 0$ is the Schwarzschild spacetime, where the cosmological horizon degenerates into $\mathscr{I}^+$. Middle: the de Sitter scenario. The cosmological horizon fixes the spacetime length scale $\lambda = r_\Lambda$. The cosmological horizon ${\cal C}^+$ and the singularity are fixed, but the event horizon is free. The corresponding limit $\eta \rightarrow 0$ is the de Sitter spacetime, where the event horizon tends to the surface $r=0$. Bottom: the Nariai scenario. The cosmological and event horizons ${\cal H}^+$ are fixed, but the singularity moves freely. The spacetime length scale incorporates a singular behavior $\lambda = r_H/(1-\eta)$, but the corresponding limit $\eta \rightarrow 1$ is finite into the Nariai spacetime.
• Figure 2: The conformal potential $\bar{V}_{\ell} ( \sigma )$ develops strong gradients around the horizons as $\eta \rightarrow 0$. In the Schwarzschild scenario (left), the strong gradient develops around the cosmological horizon as $\sigma_\Lambda$ approaches future null infinity. In the de Sitter scenario (right), it develops around the event horizon as $\sigma_h$ approaches the singularity. Insets: the rate of convergence of the corresponding Chebyshev coefficients representing $\bar{V}_{\ell} ( \sigma )$ without AnMR. As $\eta \rightarrow 0$, the strong gradient near the horizons yields a loss of accuracy, which forces one to allocate much more computing power to get a proper accuracy for a QNM solver.
• Figure 3: The optimization of the AnMR parameter $\kappa$ based on the convergence rate of Chebyshev coefficients representing the conformal potential $\bar{V}_{\ell} ( \sigma )$. Both panels illustrate the case for $\eta = 1/100$. Different values of the AnMR parameter $\kappa$ are tested numerically for the best accuracy in each case. The numerical test shows the optimized values roughly as $\kappa \approx 3.0$ for the Schwarzschild scenario (left). For the de Sitter scenario (right), the best value $\kappa \approx 2.3$ is qualitatively close $\kappa =3$, which allows one to assume the generic scaling \ref{['eq:bestAnMR']}.
• Figure 4: Top: light ring and the de Sitter QNM modes resolved without AnMR, under numerical resolutions of $N=100$ (light blue) and $N=300$ (dark blue). A noise branch in a V shape emanates from given critical value at the imaginary axis. Reliable data only exist under the noise branch, which increases with numerical resolution. Bottom: comparison of QNM values with or without AnMR (red and blue, respectively), for numerical resolution $N=300$. The noise branch changes its shape but the offset remains at same order of magnitudes. The AnMR technique allows us to calculate the light ring modes more accurately and we also find further values of light ring modes beyond the noise branch. The de Sitter modes are still valid only below the noise branch, but their evaluation is more precise, see Fig. \ref{['fig:conv_test']}.
• Figure 5: The convergence tests for QNMs, with dot markers representing the fundamental mode $n=0$, while plus markers denote the first overtone $n=1$. Color codes are kept for each value of $\eta$ across panels. Top: converge results for light ring modes without AnMR. Though exponential, the convergence rate reduces significantly as $\eta\rightarrow 0$. Middle: with AnMR enabled, the convergence rate increases and accurate results for smaller values $\eta$ are obtained with small numerical resources. Bottom: AnMR also enhances the convergence rate for de Sitter modes in the limit $\eta \rightarrow 0$.