Peaking into the abyss: Characterizing the merger of equatorial-eccentric-geodesic plunges in rotating black holes

TL;DR

This work investigates gravitational waves from critical equatorial plunges of a test particle into a Kerr black hole in the small-mass-ratio limit, using a time-domain Teukolsky framework to source waveforms from geodesics anchored by a UCO (unstable circular orbit). By spanning Kerr spins $-0.99 \le a \le 0.99$ and critical eccentricities $e_c$, the authors quantify when the dominant $h_{22}$ mode exhibits a peak and characterize merger quantities at the peak, revealing a spin- and eccentricity-dependent threshold $e_c^{\rm thr}(a)$ and an asymptotic eccentricity $e_{\rm asym}$. They extend the analysis to higher-order modes and to Bondi news $N$ and Weyl scalar $\psi_4$, showing how peak behavior propagates through the mode hierarchy and how mode-mixing and relativistic effects influence merger features. The results provide a parameter-space map of peak existence and offer a model-independent methodology to anchor merger-ringdown modeling in the small-mass-ratio, eccentric Kerr TM limit, with implications for refining eccentric waveform models and their attachment to ringdown in gravitational-wave data analysis.

Abstract

We study the gravitational waveforms generated by critical, equatorial plunging geodesics of the Kerr metric that start from an unstable-circular-orbit, which describe the test-mass limit of spin-aligned eccentric black-hole mergers. The waveforms are generated employing a time-domain Teukolsky code. We span different values of the Kerr spin $-0.99 \le a \le 0.99 $ and of the critical eccentricity $e_c$, for bound ($0 \le e_c<1$) and unbound plunges ($e_c \ge 1$). We find that, contrary to expectations, the waveform modes $h_{\ell m}$ do not always manifest a peak for high eccentricities or spins. In case of the dominant $h_{22}$ mode, we determine the precise region of the parameter space in which its peak exists. In this region, we provide a characterization of the merger quantities of the $h_{22}$ mode and of the higher-order modes, providing the merger structure of the equatorial eccentric plunges of the Kerr spacetime in the test-mass limit.

Figures (16)

• Figure 1: In the upper panel we show two critical plunge geodesics for a Kerr BH with spin $a=0$. This class of geodesics is characterized by the value of the critical eccentricity $e_c$, as described in Eq. \ref{['eq: e_LSO def']}. In the figure we show two geodesics with $e_c=\{ 0.4, \ 0.7 \}$. The bottom panel shows the radial potentials $V(r)$ of the two geodesics and their energies (horizontal lines), which correspond to the energies $\mathcal{E}_c = \{ 0.95130, \ 0.96957 \}$ of their respective UCOs. We also mark the positions of the critical radii $r_c = \{4.859, \ 4.355 \}$ and of the outer turning point $r_1 = \{ 11.3, \ 24.7 \}$ with vertical dotted lines. The vertical gray area is the area inside the BH horizon.
• Figure 2: In this figure we show the gravitational-waveform amplitudes of the $h_{22}$ mode generated by equatorial critical plunge geodesics of a Kerr BH. The left column specializes to the non-spinning case ($a = 0$), while the right column to the case with spin $a = 0.7$. The upper panels show the amplitude of the mode while the bottom panels show its first time derivative. The different colors represent different critical eccentricities $e_c$, ranging from $e_c = 0$ to $e_c = 0.9$. The peak amplitude $|h_{22}^{\rm peak}|$ and the time of peak amplitude $t_{22}^{\rm peak}$ exhibit a dependence on $e_c$. We find that in the case $a=0.7$ the peak disappears starting from a certain threshold eccentricity $e^{\rm thr}_c \approx 0.5$.
• Figure 3: This figure summarizes the two different methodologies employed to extract $e^{\rm thr}_c$ and $e_{\rm asym}$, which are the threshold eccentricities from which the peak of the amplitude of the $h_{22}$ mode disappears. In the figure we consider configurations with spin $a = 0.7$. The upper panel shows the Method 1, mentioned in Sec. \ref{['Subsec: Waveforms characterization']}: the red curve is the fit of $r_{22}^{\rm peak}$ as function of $e_c$, while the black curve is $r_c$, the radius of the UCO. The threshold eccentricity $e^{\rm thr}_c$ corresponds to the eccentricity for which the two curves intersect. We highlight the value of $e^{\rm thr}_c$ with a dashed vertical black line. The bottom panel shows the Method 2: the magenta dots represent the quantity $\Delta t_{22}$, which exhibits a logarithmic divergence at $e_{\rm asym}$. After performing a fit of $\Delta t_{22}$ (magenta curve), we extract the value of the asymptote $e_{\rm asym}$ (dot-dashed vertical black line) and we compare it with $e^{\rm thr}_c$.
• Figure 4: Characterization over the parameter space $(a, e_c)$ of the peak existence of the $h_{22}$ mode amplitude. The blue area is the part of parameter space for which the peak exists while the red area is the part for which the peak does not exist. The black dots are the values of $e_c^{\rm thr}(a)$ computed with Method 1 as explained in Sec. \ref{['Subsec: Waveforms characterization']}. The blue curve is a quadratic polynomial fit of $e_c^{\rm thr}$ as function of $a$. The inset shows a zoom of $e_c^{\rm thr}(a)$ for values of the spin $0.85 \le a \le 1$ and critical eccentricity $0 \le e_c \le 1.2$. We find that for $a \ge 0.95$, it is impossible to find the peak of the amplitude of the $h_{22}$ mode even when $e_c = 0$.
• Figure 5: In the top panel of this figure we show the amplitude of the $h_{22}^{\rm N}$ mode defined in Eq. \ref{['eq.:h22N expr']}, which corresponds to the Newtonian order of the $h_{22}$ mode. We compute the quantity $|h_{22}^{\rm N}|$ on critical plunge geodesics with spin $a = 0.7$ and values of critical eccentricity $e_c \le 1$, and we plot it as a function of the radial coordinate $r$. The bottom panel shows the quantity $r \Omega$ of the considered critical plunge geodesics as a function of $r$. The different curves start at different radii because of the fact that the critical plunge geodesics asymptotically start at the critical radius $r_c$, which is a function of the quantities $a$ and $e_c$, as mentioned in Sec. \ref{['Sec.:critical plunge geodesics']}.