Coexistence of Spectrally Stable and Unstable Modes in Black Hole Ringdowns

TL;DR

The paper addresses spectral instability in black hole quasinormal modes (QNMs) that arises when a secondary potential barrier forms, producing a coexisting off-peak family alongside the traditional peak (photon-sphere) modes. Using a static, spherically symmetric Einstein-Maxwell-scalar (EMS) hairy black-hole background and a massless scalar perturbation, the authors combine spectral methods and time-domain simulations to analyze both frequency-domain QNMs and the corresponding ringdown signals. They identify two QNM families, peak and off-peak, whose coexistence persists even after the potential well vanishes, due to a residual scale from the valley; yet the time-domain signal remains dominated by the spectrally stable peak family, with off-peak modes contributing only subdominantly. The results reinforce the robustness of black hole spectroscopy, showing that observable ringdown is largely determined by the more spectrally stable modes, and provide a self-consistent mechanism explaining why spectral instability does not necessarily threaten the interpretation of early-time gravitational-wave signals.

Abstract

Recent studies have shown that a secondary potential barrier, forming a potential well outside the event horizon, can destabilize the Quasinormal Mode (QNM) spectrum of black holes. We find that spectral instability may persist even after the potential well vanishes, giving rise to a distinct family of spectrally unstable QNMs that differ from the spectrally stable modes localized near the potential peak and associated with the photon sphere. Nevertheless, time-domain simulations reveal that early-time ringdown waveforms remain dominated by stable modes, while unstable modes have only a subdominant contribution. These results highlight the robustness of black hole spectroscopy, as the observable ringdown signal is primarily governed by the most stable QNMs.

Figures (6)

• Figure 1: QNM spectrum of the scalar field with $l=10$ for hairy BHs with varying charge-to-mass ratio $Q/M$. Diamonds denote the peak modes $\omega_{n}^{(p)}$, indexed by $n_{p}$, while plus signs indicate the off-peak modes $\omega_{n}^{(o)}$, indexed by $n_{o}$. Left:Trajectories of QNM frequencies in the complex plane, illustrating how the modes migrate as $Q/M$ varies. The color bar indicates $Q/M$. Diamonds denote the peak modes, indexed by $n_{p}$, while plus signs indicate the off-peak modes, indexed by $n_{o}$. Left:Trajectories of QNM frequencies in the complex plane, illustrating how the modes migrate as $Q/M$ varies. Blue triangles mark the transition points separating the single-peaked and double-peaked regimes of the effective potential $V_{\text{eff}}$; the portions of the trajectories below and to the left of the triangles correspond to the single-peaked regime. In this regime, the off-peak modes exhibit much larger frequency migrations than the peak modes, indicating that the latter are more spectrally stable under variations in $Q/M$. Upper right: Square of the real part of the QNM frequency, $\omega_{R}^{2}$, as a function of $Q/M$, together with representative profiles of $V_{\text{eff}}$. The highest peak of $V_{\text{eff}}$ corresponds to each selected $Q/M$. The cyan-shaded region indicates the double-peaked regime. The peak modes are associated with oscillations near the potential peak (photon-sphere modes) in the single-peaked regime, whereas the low-lying off-peak modes correspond to states trapped within the potential valley in the double-peaked regime. Lower right: Imaginary part of the QNM frequencies as a function of $Q/M$. A fundamental-mode overtaking occurs between the peak and off-peak families. The fundamental QNM is identified as the slowest-decaying mode among peak and off-peak families in the white- and green-shaded regions, respectively. Red triangles mark the overtaking event in the complex-frequency plane.
• Figure 2: QNM spectrum of the scalar field with $l=2$ for hairy BHs with varying charge-to-mass ratio $Q/M$. Compared with the $l=10$ case, the $l=2$ spectrum exhibits a much stronger influence from the remnant of the double-peaked structure within the single-peaked regime. This enhanced residual effect renders the $n_{p}=2$ overtone of the peak-mode family spectrally unstable, while the $n_{p}=0$ and $n_{p}=1$ modes remain spectrally stable provided $Q/M$ stays sufficiently far from the double-peaked regime.
• Figure 3: QNM fitting results for $Q/M=1.04436$ with $l=10$. The waveform is fitted using the strong fit model with three QNMs and the weak fit model with $\left( N_{p},N_{o}\right) =\left( 3,1\right)$. Left: QNM frequencies extracted from the strong fit model in the complex plane for varying start times, represented by shaded dots. The color bar indicates the start time. Diamonds and plus signs denote the frequencies of the peak and off-peak families computed in the frequency domain, respectively. Only the peak modes can be reliably extracted. Upper right: Extracted amplitudes $A_{n}$ from the strong fit model as functions of the start time $t_{0}$. Lower right: Amplitudes obtained from the weak fit model including the first three peak modes and one off-peak mode. The off-peak mode amplitude is approximately $\mathcal{O}\left( 10^{-4}\right)$ smaller than those of the peak modes, demonstrating that the off-peak contribution to the time-domain waveform is negligible.
• Figure 4: QNM fitting results for $l=2$ with $Q/M=1.03507$. The waveform is fitted using the strong fit model with two QNMs and the weak fit model with $\left( N_{p},N_{o}\right) =\left( 2,2\right)$. Left: QNM frequencies extracted from the strong fit model in the complex plane. Dots and crosses correspond to early and late fitting windows, respectively. For early windows, only the peak modes are recovered, indicating that the early-time waveform is dominated by these modes. For late windows, both the $n_{p}=0$ peak and $n_{o}=0$ off-peak modes are extracted, as they are the slowest-decaying QNMs. Upper right: Extracted amplitudes $A_{n}$ from the strong fit model as functions of the start time $t_{0}$ for early and late fitting windows. Lower right: Amplitudes obtained from the weak fit model. The off-peak mode amplitudes are approximately $\mathcal{O}\left( 10^{-3}\right) -\mathcal{O}\left( 10^{-1}\right)$ smaller than those of the peak modes, confirming their subdominant contribution to the time-domain waveform.
• Figure 5: Migration rates $\delta\omega _{R}$ (Left) and $\delta\omega_{I}$ (Right) for the real and imaginary parts of the QNM frequencies, shown as a function of $Q/M$. The upper and lower rows correspond to the $l=10$ and $l=2$ cases, respectively.