Ringdown modeling for effective-one-body waveforms in the test-mass limit for eccentric equatorial orbits around a Kerr black hole

Abstract

We study the plunge and merger of a non-spinning particle falling into a Kerr black hole following an eccentric planar inspiral. The dynamics is driven by an effective-one-body radiation reaction, and the corresponding numerical inspiral-merger-ringdown waveforms are obtained by solving the Teukolsky equation with the 2+1 time-domain code Teukode. We then analyze in detail the plunge and merger phases, modeling the merger-ringdown waveform using closed-form ansätze. Crucially, our modeling starts from a point closely related to the light-ring crossing, rather than from the amplitude peaks. This choice allows us to neglect the impact of the relativistic anomaly at the separatrix-crossing, and to extend the modeling to high spins and high eccentricities. We model all the multipoles with $m\geq 1$ up to $\ell=4$, as well as the $(2,0)$, $(5,5)$, $(5,4)$, and $(5,3)$ modes, including spherical-spheroidal mode-mixing and the beating between co-rotating and counter-rotating quasi-normal modes. The post-merger waveform model is then employed to complete an effective-one-body inspiral-plunge waveform, thus providing a complete description. Our model, built using elliptic-like configurations for the merger-ringdown phase, naturally extends to dynamical capture scenarios without any further modification. Finally, we provide insights into the extension of this framework to generic mass ratios, arguing that a time closely related to the inflection point of the (2,2) waveform frequency could be used as anchoring point for the ringdown modeling.

Figures (23)

• Figure 1: Left panel: amplitudes $A_{22}$ (warm colors) and frequencies $\omega_{22}$ (cold colors) for RWZ-normalized waveforms $\Psi_{22}$ for quasi-circular systems with different Kerr spins, ${\hat{a}}\in[-0.7,0.7]$ (reported in the legend). Dots on the amplitudes mark the locations of the maxima, ${t_{A_{22}}^{\rm peak}}$, while dots on the frequencies mark the inflection point, ${ t_{\dot{\omega}_{22}}^{\rm max}}$. Waveforms aligned with respect to the lr crossing ${t_{\rm LR}}$. Middle panel: same quantities, but for ${\hat{a}}=-0.6$ and eccentricities ${e_{s}}\in[0,0.9]$. Right panel: as the middle panel, but for ${\hat{a}}=0.6$.
• Figure 2: Potentials $V(r;p_\varphi)$ for three cases with ${\hat{a}}=0.3$, ${e_{s}}=0.5$ but different anomalies at the separatrix crossing: ${\xi_{s}} = \left( 0, \pi, 3.20 \right)$. We show the potential at the apastron passages (blue/green solid lines), at ${t_{s}}$ (dashed orange), and at ${t_{\ddot{r}=0}}$ (dash-dotted purple). The orange cross marks the radial location of the test-mass at ${t_{s}}$, and the purple plus marks it at ${t_{\ddot{r}=0}}$. The same markers are used on the radial evolution $r(t)$ reported in the bottom panels, together with circles which mark the apastron (same color scheme of the potentials).
• Figure 3: Upper panel: Teukode amplitudes of the (2,2) mode for configurations with ${\hat{a}}=0$, ${e_{s}}=0.5$, $\nu=10^{-3}$, but different ${\xi_{s}}$ (reported in the legend), obtained with $N_r\times N_\theta=5401\times 321$ resolution. The dots mark ${t_{s}}$. Bottom left: corresponding eccentricities as a function of time. Bottom right: relative difference of $A_{22}$ and $\omega_{22}$ between the configurations with ${\xi_{s}}>0$ and the one with ${\xi_{s}}=0$. These differences are also shown in purple in Fig. \ref{['fig:anomaly_summary']}.
• Figure 4: Relative differences for the amplitude and frequency of the (2,2) mode between configurations with ${\xi_{s}}>0$ and the one with ${\xi_{s}}=0$ computed at ${t_{A_{22}}^{\rm peak}}$ (hollow tick markers) and at ${t_{\rm LR}}$ (faint filled markers). Different colors refer to different $({\hat{a}},{e_{s}},\nu)$-configurations, see legend.
• Figure 5: Amplitude (upper panel) and frequency (lower panel) of the (2,2) mode for ${\hat{a}}=0.9$, ${e_{s}}=0.5$ and different ${\xi_{s}}$, shifted with respect to the lr crossing (vertical line). The dots mark the amplitude maxima and the inflection points of the frequency. The amplitudes aligned with respect to ${t_{A_{22}}^{\rm peak}}$ rather than ${t_{\rm LR}}$ are shown in Fig. \ref{['fig:anomaly_kerr']}.