Black hole quasinormal mode resonances

Abstract

Black hole quasinormal mode frequencies can be very close to each other ("avoided crossings") or even completely degenerate ("exceptional points") when the system is characterized by more than one parameter. We investigate this resonant behavior and demonstrate that near exceptional points, the two modes are just different covers of the same complex function on a Riemann surface. We also study the characteristic time domain signal due to the resonance in the frequency domain, illustrating the analogy between black hole signals at resonance and harmonic oscillators driven by a resonant external force. We consider a specific toy model displaying a resonance between the fundamental mode and the first overtone, and we find that taking into account the linear growth in time due to the resonance is necessary to accurately recover the quasinormal mode frequencies.

Figures (4)

• Figure 1: Schematic representation of Riemann surfaces and avoided crossings. Since the main features are universal, for simplicity we use the two-level system toy model described in the Supplemental Material SM_Ref. The system contains two free parameters denoted by $p_1$ and $p_2$, corresponding to the real part and imaginary part of a complex variable $z$. In the two central panels we show how the real and imaginary parts of the eigenvalues change as we complete a loop in the parameter space (leftmost panel): after one loop around the EP we do not recover the original eigenvalue. The blue line in the leftmost panel shows a characteristic path for eigenvalue repulsion, and the corresponding $\delta\omega_{\pm}=\omega_{+}-\omega_{-}$ is shown by the red line in the rightmost panel. The black dashed line is a hyperbola, that fits the trajectory of $\delta\omega_{\pm}$ very well Motohashi:2024fwt.
• Figure 2: Riemann surface structure of the QNMs around the EP for the bump potential in Eq. (\ref{['bump']}). As in Fig. \ref{['fig:riemannsurface']}, the real part and the imaginary part are displayed separately. The resonance between the fundamental mode and the first overtone occurs at $\epsilon_{\star}=10^{-2.294}\approx0.005$ and $d_{\star}=15.698$, where the modes coalesce into $\omega_{\star}\approx0.365-0.117\mathrm{i}$.
• Figure 3: Time-domain waveforms and fitting results in three different cases. (a) Upper left: ringdown signal for the unperturbed RW potential (black), a perturbative bump with $\epsilon=0.05\approx10\epsilon_{\star}$ and $d=d_{\star}$ (red), and for bump parameters fine tuned to resonance (green). We have chosen the time origin $t=0$ (somewhat arbitrarily) to exclude the initial transient part and focus on the ringdown. The vertical dashed lines are the boundaries of the fitting region, which ranges from $t=30$ ($\tau_\mathrm{fit}=0$) to $t=110$. (b) Upper right: mismatch as a function of the starting time of the fit, $\tau_{\text{fit}}=t_{\mathrm{start}}-30$. Solid lines refer to the waveforms in panel (a), fitted using a single QNM waveform; the green dashed line refers to a fit of the resonant waveform with a resonant ringdown signal. Bottom panels: frequencies inferred by fitting the corresponding signals. The red cross marks the fundamental mode. Colored crosses/circles are the frequencies found by varying the starting time of the fit from $t_{\mathrm{start}}=30$ or $\tau_{\text{fit}}=0$ (light blue) to around $t_{\mathrm{start}}=75$ or $\tau_{\text{fit}}=45$ (purple) for the RW potential in panel (c), the non-resonant bump in panel (d), and the resonant bump in panel (e). In panel (e), the crosses are frequencies inferred by fitting a single QNM to the resonant signal, while the circles were found by fitting a resonant waveform model.
• Figure 4: Results of fitting a resonant waveform $\psi$ using resonant and non-resonant waveforms with $n=4$. The red line is the residual $\Delta h_{\text{DS}}=\psi-h_{\text{DS}}(t)$ between the resonant waveform and a model with only damped sinusoids, while the blue line is the corresponding residual $\Delta h_{\text{R}}=\psi-h_{\text{R}}(t)$ for the resonant waveform model. The black dashed line is the result of fitting $\Delta h_{\text{DS}}$ again with $h_{\text{R}}(t)$.