Quasinormal ringing of Kerr black holes. III. Excitation coefficients for equatorial inspirals from the innermost stable circular orbit

TL;DR

This paper develops a frequency-domain perturbative framework to compute quasinormal mode excitation coefficients $C_q$ for equatorial plunges from the innermost stable circular orbit in Kerr spacetimes. By employing the Teukolsky equation, the Sasaki-Nakamura transformation, and a regularized Green's-function approach, the authors extract the SN excitation coefficients $\tilde{C}_q^{\rm SN}$ and relate them to the gravitational-strain amplitudes $C_q$ that govern the ringdown signal. The study reveals a strong spin dependence: while the fundamental $\ell=m=2$ mode dominates at modest spins, higher overtones and higher multipoles become increasingly excited as the black hole spins approach extremality, potentially enhancing their detectability in high-SNR observations (e.g., LISA). The results provide a quantitative calibration of ringdown amplitudes in the extreme mass-ratio limit and offer guidance for incorporating overtones into waveform models for accurate parameter estimation in highly spinning mergers.

Abstract

The remnant of a black hole binary merger settles into a stationary configuration by "ringing down" through the emission of gravitational waves that consist of a superposition of damped exponentials with discrete complex frequencies - the remnant black hole's quasinormal modes. While the frequencies themselves depend solely on the mass and spin of the remnant, the mode amplitudes depend on the merger dynamics. We investigate quasinormal mode excitation by a point particle plunging from the innermost stable circular orbit of a Kerr black hole. Our formalism is general, but we focus on computing the quasinormal mode excitation coefficients in the frequency domain for equatorial orbits, and we analyze their dependence on the remnant black hole spin. We find that higher overtones and subdominant multipoles of the radiation become increasingly significant for rapidly rotating black holes. This suggests that the prospects for detecting overtones and higher-order modes are considerably enhanced for highly spinning merger remnants.

Figures (14)

• Figure 1: Integration contour to isolate the QNM contribution Berti:2006wq. The contour is closed in the lower semi-half including QNMs marked by crosses. The shaded are is the branch cut, whose contribution to the waveform is not discussed here.
• Figure 2: Comparison between the real part of the full SN function \ref{['eq:wf_SN_complete']} (thick solid black line) and of the mode superposition \ref{['eq:mode_expansion']} including QNMs up to the overtone number $n$ shown in the legend (thin colored lines). We assume $a/M=0.68$ and $\ell =m=2$. Here, $u_{\rm LR}$ is the position of the light ring in terms of the null coordinate $u=t-r_\star$.
• Figure 3: Excitation coefficients for $\ell = m = 2$ and different overtone numbers $n$ as functions of $a/M$. The solid blue, dashed red and dotted green lines refer to the fundamental mode ($n=0$), the first overtone ($n=1$) and the second overtone ($n=2$), respectively. The main (top) panel shows the absolute value (phase) of the excitation coefficients. The right panel is a zoomed-in view of the absolute value in the region $a/M > 0.9$, highlighted by a red rectangle in the main panel. This close-up reveals that for rapidly rotating BHs ($a/M \gtrsim 0.994$) the fundamental mode does not correspond to the largest value of $|C_q|$.
• Figure 4: Excitation coefficient trajectories, parametrized by $a/M$, for $\ell = m = 2$ and different overtone numbers. The solid blue, dashed red and dotted green lines refer to the fundamental mode ($n=0$), the first overtone ($n=1$) and the second overtone ($n=2$), respectively. Thick lines highlight the near-extremal regime $a/M > 0.9$, while markers indicate selected spin values, as identified in the legend.
• Figure 5: Same as Figure \ref{['fig:exc_coeff_main']}, but for $\ell=m=2,\,3,\,4$ and $n=0,\,1$. The legend shows the pair $(\ell=m,\,n)$. Lines in different shades of blue (red) correspond to $n=0$ ($n=1$). We observe that $|C_{220}|$ is the largest for $a/M\lesssim0.994$, while the values of $|C_q|$ for all modes with $n=1$ become larger than $|C_{220}|$ close to extremality.