Resonance in black hole ringdown: Benchmarking quasinormal mode excitation and extraction

Abstract

We investigate how resonant excitation near exceptional points manifests in Kerr black hole ringdown waveforms and examine its extraction. Using waveforms generated by localized initial data, where quasinormal mode amplitudes are given solely by excitation factors, we establish a controlled benchmark for overtone extraction. Applying an iterative fitting method with mirror modes, we analyze a mild resonance in the $(l,m)=(2,2)$ multipole and a sharp resonance in the $(3,1)$ multipole occurring as part of a sequence of successive resonances. For $(2,2)$, we extract the fundamental mode, the first three overtones, and the fundamental mirror mode with relative errors below $10\%$, and show that residual waveforms exhibit the expected damped sinusoids together with distinctive resonance signatures. For $(3,1)$, we demonstrate that resonances can not only amplify but also reduce quasinormal mode excitations, reshaping the overtone hierarchy and rendering the sharp resonance more pronounced in ringdown. Our results clarify the imprint of resonance in Kerr ringdown and highlight both the robustness and limitations of current extraction techniques, providing a foundation for more reliable extraction of higher overtones and for applications to observational data analysis.

Figures (15)

• Figure 1: Integration contour for Eq. \ref{['eq:Psi4atinfinity']}. The cross markers and the jagged line denote the QNM poles and the branch cut, respectively. The integral along the real frequency axis in Eq. \ref{['eq:Psi4atinfinity']} is equivalent to closing the contour in the lower half-plane, provided that the contributions from the branch cut and the semicircular arc at infinity are properly taken into account. By the residue theorem, the integral over the closed contour yields the sum over QNMs, while the integral along the branch cut gives rise to the power-law tail. The contribution from the semicircular arc vanishes by Jordan's lemma.
• Figure 2: QNM frequencies for $(l,m)=(2,2)$
• Figure 3: Excitation factor for $(l,m)=(2,2)$
• Figure 4: QNM frequencies for $(l,m)=(3,1)$
• Figure 5: Excitation factor for $(l,m)=(3,1)$