QNM families: classification and competition

TL;DR

This work shows that a hairy Schwarzschild black hole with a double-peak perturbative potential supports two distinct QNM families, the photon-sphere $PS$ modes and long-lived echo modes, whose dominance in the ringdown is dynamically governed and highly sensitive to the initial perturbation. By combining frequency-domain classification via WKB and pseudospectral methods with a time-domain windowed energy analysis, the authors demonstrate that long-lived echo modes can dominate within finite observation windows, even when PS modes appear dominant in the stationary spectrum. The study reveals that the spectral hierarchy reorganizes across the echo parameter space and that the perturbation source leaves a lasting imprint on which modes are excited and how energy is partitioned over time. These results provide a novel observational handle on beyond-GR physics through ringdown signals and motivate targeted analyses of long-lived modes in gravitational-wave data.

Abstract

The perturbation spectra of black hole (BH) beyond standard general relativity (GR) may exhibit complex structures with long-lived modes that generate echo-like modulations on the ringdown signal. However, a systematic framework for understanding the internal structure of such spectra, the physical nature of different mode families, and their dynamical excitation is still undeveloped. In this paper, we address this issue by proposing a potential methodology that combines frequency-domain classification with time-domain analysis, using a hairy Schwarzschild BH that admits a double-peak perturbative potential as a theoretical platform. Our analysis of the quasinormal mode (QNM) spectrum identifies two distinct families of modes: the photon sphere (PS) family, arising from delocalized scattering resonances, and the echo family, corresponding to highly localized quasi-bound states. We then develop a windowed energy analysis framework in the time domain, which discloses a dynamic competition for dominance between these families. In particular, our results explicitly show that this competition is sensitive to the properties of the initial perturbation source, and that higher-overtone echo modes can dominate in the observed signal, which are in contrast to the standard PS mode in GR. This study establishes the dynamic evolution of this energy competition as a new observational signature for probing new physics and further motivate a supplemental framework for analyzing long-lived ringdown signals.

Figures (4)

• Figure 1: (a): Parameter spaces of the effective potential morphology for electromagnetic perturbations with $s=1$, $\ell=6$ and $M=1$. The grey shaded region indicates the $\{\alpha, l_0\}$ parameter space where the potential exhibits a double-peak structure, a prerequisite for echoes. The blank region corresponds to a typical single-peak potential. The black curve shows the constraint $l_0=2M\alpha/(e^2+\alpha)$ where the hairy black hole's horizon coincides with the seed Schwarzschild radius. The horizontal and vertical red dashed lines mark the potential varying along typical parameter paths analyzed in (b) and (c), respectively. (b): Typical potential profiles along the horizontal path from (a) at a fixed $\alpha=7$. This path demonstrates the transition from the Outer-Single-Peak regime (left end) to the Inner-Single-Peak regime (right end) as $\alpha$ increases. (c): Typical potential profiles along the vertical path from (a) at a fixed $l_0/M=0.94$. This path demonstrates the transition from the Inner-Single-Peak regime (bottom end) to the Outer-Single-Peak regime (top end) as $l_0$ increases.
• Figure 2: Top Panel: The imaginary (left) and real (right) parts of the QNM spectra as $l_0/M$ is varied at fixed $\alpha=7$. This corresponds to a vertical traversal of the parameter space shown in Fig. \ref{['fig:peak-number-phase-diagram']}. Bottom Panel: The imaginary (left) and real (right) parts of the QNM spectra as $\alpha$ is varied at a fixed $l_0/M=0.94$. This corresponds to a horizontal traversal. In each panel, the vertical red dashed lines mark the boundaries between the single-peak and double-peak (echo) regions. Blue Dots: High-precision data from the pseudospectral method, filtered by a spectral convergence test (drift ratio $\ge 10^3$Chen:2024monBOYD199611). Colored Polygons: The squares, triangles, and other polygons represent the first mode estimated by WKB approximation obtained from different extrema of the effective potential. They serve to classify the pseudospectral method results into: the PS family (OPS mode for the Outer-Single-Peak, IPS mode for the Inner-Single-Peak) and the echo family (associated with the potential well). Purple Diamonds: These points are calculated using the direct integration method. They provide an independent validation for the imaginary part of the long-lived echo modes, where the WKB approximation is not applicable. The excellent agreement confirms the accuracy of our pseudospectral method results for the damping rates of these modes.
• Figure 3: Left Panel: In the Inner-Single-Peak region, the spacial configuration of the fundamental mode (with $M\omega_0=1.405711-0.033302~ \mathrm{i}$) is significantly condensed near the potential's "shoulder", demonstrating the diffraction trapping effect. Middle Panel: In the double-peak region, the spacial configuration of the fundamental mode (with $M\omega_0=1.121228-1.569230\times 10^{-7} ~ \mathrm{i}$) is highly localized within the potential well, exhibiting a clear quasi-bound state (QBS) character. Right Panel: In the Outer-Single-Peak region, the spacial configuration of the fundamental mode (with $M\omega_0=1.246082-0.096051~ \mathrm{i}$) is highly delocalized, which is a characteristic of a standard scattering resonance associated with the PS. In each panel, the black dashed lines indicate the energy level, $\text{Re}(\omega_n)^2$, of the corresponding QNM from the pseudospectral data.
• Figure 7: The functions $\gamma^{\text{in-out}}(\sigma)$, $\tilde{\gamma}(\sigma)$ and $\gamma^{\text{out-in}}(\sigma)$ are shown. For the left panel with the in-out strategy, the parameters are chosen as $M=1$, $l_0=1.02M$ and $\alpha=9.312$. For the right panel with the out-in strategy, the parameters are chosen as $M=1$, $l_0=1.02M$ and $\alpha=7.312$. The above parameter selections are from Yang:2024rms.