Quasinormal modes of Schwarzschild-de Sitter black holes in semi-open systems

TL;DR

The paper investigates perturbations of Schwarzschild-de Sitter black holes in a semi-open system created by a partially reflecting wall near the horizon. It develops an exact analytic framework using Heun functions to solve the perturbation equations and derives a QNM condition in the presence of a frequency-dependent reflectivity $\mathcal{K}(\omega)$. The study reveals three distinct regimes of QNM migration as the (real) reflectivity $\mathcal{K}$ increases, analyzes how greybody factors exhibit resonant oscillations controlled by the effective cavity length, and demonstrates the emergence of a second-order exceptional point when $\mathcal{K}$ is extended to the complex plane, accompanied by mode exchange and Puiseux-sqrt scaling near the EP. These results provide a rigorous link between horizon reflectivity models and observable ringdown and Hawking-like spectra, with implications for exotic compact objects and future extensions to rotating spacetimes.

Abstract

We study perturbations of Schwarzschild-de Sitter black holes in semi-open systems by using the Heun functions. For the semi-open system, a partially reflective wall is added around the event horizon. Three aspects of this model are investigated, namely the quasinormal mode (QNM) spectra, the greybody factor (GF), and the exceptional point (EP). For the QNM aspect, we identify three distinct behaviors as the frequency-independent reflectivity $\mathcal{K}$ increasing. The first-type modes approach the real axis and form long-lived quasi-bound states. The second-type modes move toward but do not reach the real axis and retain a finite decay rate. The third-type modes eventually lie on the imaginary axis becoming purely decaying modes. For the GF aspect, GFs exhibit strong oscillations controlled by the distance between the potential and the reflective wall with a real constant reflectivity. In contrast, a Boltzmann-type reflectivity produces only small corrections. Finally, by promoting $\mathcal{K}$ to a complex parameter, the modified boundary conditions give rise to a second-order EP. Parameterizing the vicinity of such EP, we observe the mode exchange phenomenon, and the deviation of spectra scale with the square root of the deviation of the parameter, as predicted by a Puiseux series expansion.

Figures (8)

• Figure 1: The migrations of the QNM spectra with $\mathcal{K}\in[0,1]$ varying in the complex plane for the first $10$ modes, where the parameters of left panel are chosen $r_e=0$, $r_c=1.1$, and $x_0=-100$ while the parameters of right panel are chosen $r_e=0$, $r_c=1.1$, and $x_0=-150$. Different color lines correspond to different modes. Dots of different shapes represent $\mathcal{K}$ of different intensities, where $\star=0$, $\otimes=10^{-60}$, $\oplus=10^{-56}$, $\blacklozenge=10^{-52}$, $\blacktriangle=10^{-48}$, $\blacktriangledown=10^{-44}$, $\blacksquare=10^{-40}$, $\bullet=10^{-36}$, $\spadesuit=10^{-32}$, $\heartsuit=10^{-28}$, $\clubsuit=10^{-24}$, $\diamondsuit=10^{-20}$, $\triangle=10^{-16}$, $\triangledown=10^{-12}$, $\square=10^{-8}$, $\circ=10^{-4}$, and $\ast=1$, the same below. Two subfigures in left panel are the zoom-in near the imaginary axis for the modes $n=7$ and $n=9$. The symbol $\blacktriangleright$ with pink color in the left panel stands for the parameter $\mathcal{K}=3.57\times 10^{-14}$.
• Figure 2: The migrations of the QNM spectra with $\mathcal{K}\in[0,1]$ varying in the complex plane for the first $7$ modes, where the parameters of left panel are chosen $r_e=0$, $r_c=100$, and $x_0=-20$ while the parameters of right panel are chosen $r_e=0$, $r_c=100$, and $x_0=-30$. Three subfigures above are the zoom-in near the imaginary axis.
• Figure 3: Greybody factor $\Gamma(\omega)$ for axial gravitational $(s=2)$ perturbations of the Schwarzschild-de Sitter black hole with a frequency-independent reflectivity $\mathcal{K}(\omega)$. In both panels the reflective wall is placed at $x_0=-30$, and the curves with different colors correspond to $\mathcal{K}(\omega)=0,\,0.25,\,0.50,\,0.75,$ and $1$. The left panel shows the near-extremal case with $r_e=1$ and $r_c=1.1$, and the right panel shows the nearly Schwarzschild case with $r_e=1$ and $r_c=100$.
• Figure 4: Greybody factor $\Gamma(\omega)$ for axial gravitational $(s=2)$ perturbations of the Schwarzschild-de Sitter black hole with constant reflectivity $\mathcal{K}(\omega)=0.5$ and different positions of the reflective walls. The curves in panel (a) correspond to $x_0=-150$, $-100$, and $-50$ in the near-extremal case with $r_e=1$, $r_c=1.1$. Panels (b), (c), and (d) show respectively, $r_c=2$, $r_c=10$, and $r_c=100$, with $x_0=-50$, $-30$, and $-10$ in each panel.
• Figure 5: Greybody factors $\Gamma(\omega)$ and reflection coefficients $R(\omega)$ for axial gravitational $(s=2)$ perturbations of the Schwarzschild-de Sitter black hole in the Boltzmann-type reflectivity model. Panels (a), (c), and (e) show $\Gamma(\omega)$, whereas panels (b), (d), and (f) display the corresponding reflectivity $R(\omega)$ for the same parameters. In each panel, the curve labelled $\mathcal{K}=0$ corresponds to the greybody factor of the original non-reflective SdS black hole. The top row corresponds to the near-extremal case with $r_e=1$ and $r_c=1.1$, with the reflective wall positions $x_0=-150$, $-100$, and $-50$. The middle row corresponds to $r_e=1$ and $r_c=10$ with $x_0=-50$, $-30$, and $-10$, and the bottom row to $r_e=1$ and $r_c=100$ with the same set of $x_0$.