Instability of parametrized black hole quasinormal modes in the high-overtone limit via the exact WKB analysis

TL;DR

This work analyzes the high-overtone quasinormal modes (QNMs) of parametrized black holes using the exact WKB method. By incorporating beyond-GR corrections into the Regge–Wheeler potential and validating against Leaver's numerical results, the authors show that certain parameter choices drive the real parts of high-overtone frequencies to diverge, in contrast to the GR Schwarzschild case where Re(ω_n) converges. Two concrete corrections, δQ3 and δQ4, are treated in detail: δQ3 yields a Re(ω) that can diverge as log N for specific α3 values, while δQ4 produces a slower power-law growth Re(ω) ~ N^{1/5}. The results imply that GR’s universal high-overtone convergence is not generic under beyond-GR parametrizations and highlight the nuanced interplay between turning-point geometry and Stokes phenomena in determining QNM spectra, with potential implications for quantum-gravity interpretations.

Abstract

We study the asymptotic behavior of parametrized black hole quasinormal modes (QNMs) in the high-overtone limit. To gain insights into their analytical structure, we apply the exact WKB method, which was recently developed by the same authors. Our theoretical predictions are confirmed in good agreement with the numerical results based on Leaver's method. For specific values of parametrization parameters that characterize deviations from general relativity, we find that the real part of asymptotic QNM frequencies diverges in the high-overtone limit, in sharp contrast to the case of a Schwarzschild black hole. This demonstrates that the convergence of the real parts of high-overtone QNMs is a distinctive feature of general relativity, while parametrized corrections generically lead to divergent spectral behaviors.

Figures (5)

• Figure 1: Schematic picture of the Stokes curves for the potential $Q_{0}(r)=r-r_i$ showing the branch cut and the counter-clockwise analytic continuation from the Stokes region $U_1$ to $U_2$ across the Stokes curve $\Gamma$. The triangle marker represents a turning point at $r = r_i$. The branch cut originating from $\sqrt{Q_0(r)}$ is shown as the dotted line. The dominant WKB solution, $\mathcal{S}[\psi_{+}]$ or $\mathcal{S}[\psi_{-}]$, on each Stokes curve is denoted by + or -, respectively. We use these symbols throughout this paper.
• Figure 2: Schematic picture of the Stokes curves for the potential $Q_{\text{RW},0}$ with the branch cuts (see Fig. \ref{['fig:SL_r-r_i']} for the explanation of the symbols). The open squares represent the poles of $Q_0(r)$, $r=0$ and $r=1$. For illustration, we take $l=s=2$ and $\omega = 3.0 - 3.3 i$. The curves flowing to $r = 1$ forms a logarithmic spiral.
• Figure 3: The asymptotic behavior of $\text{Re}(\omega_N)$ for the case of $\delta Q_3$. The black (dense) dots and black dashed line correspond to the numerical result (Leaver) and theoretical prediction (exact WKB), respectively. We take $\alpha_3 = 20/9$ (left) and $\alpha_3 = 32/9$ (right) with $s = l = 2$, for which $\text{Re}(\omega_N) \to \infty$ in the high-overtone limit $N \to \infty$.
• Figure 4: Schematic pictures of the Stokes curves for the potential $Q_{0}^{(4)}$ with the branch cuts. We take $l=s=2$ and $\omega = 6 - 6 i$ for both panels. The coefficient $\alpha_4$ is set to $\alpha_4=1$ (left) and $\alpha_4=-1$ (right).
• Figure 5: The asymptotic behaviors of $\text{Re}(\omega_N)$ for $\delta Q_4$ with $\alpha_4 = +1$ (left) and $\alpha_4 = -1$ (right) are shown, where we take $s = l = 2$. The black dense dots and black dashed line correspond to the numerical result (Leaver) and the prediction of the exact WKB analysis, respectively.

Theorems & Definitions (1)

• Theorem 2.2