Modeling the merger-ringdown of an eccentric test-mass inspiral into a Kerr black hole using the effective-one-body framework

Abstract

We characterize and phenomenologically model the merger-ringdown of gravitational waves emitted by a small compact object that plunges and merges into a Kerr black hole from equatorial-eccentric inspirals. The waveforms are generated employing a time-domain Teukolsky code sourced with trajectories computed using the effective-one-body framework. We span values of the Kerr spin $a\in[-0.9, 0.9] $, eccentricity at the last stable orbit (LSO) $ e_{\rm LSO} \in [0,0.9] $, and relativistic anomaly $ ξ_{\rm LSO} \in [0 , 2 π]$. We characterize the last peak of the waveform and ringdown features across the parameter space, finding that the eccentricity mainly affects the last peak features, while it has a smaller impact on the ringdown signal. In contrast, the relativistic anomaly measured at the LSO influences the morphology of the last peak in a restricted portion of the parameter space and has no impact on the ringdown part. We perform the analysis for all the spin-weighted spherical harmonic modes normally included in the \texttt{SEOBNR} family of models, $(\ell,m)\in\{ (2,2), (3,3), (4,4), (5,5), (2,1), (3,2), (4,3)\}$. Finally, we introduce a merger-ringdown model for \texttt{SEOB-TMLE}, a forthcoming inspiral-merger-ringdown waveform model for eccentric spin-aligned binary black holes in the test-mass limit, whose features can be extended to comparable-mass regimes. The model also accounts for quasinormal mode mixing during the ringdown. It provides a first step toward incorporating the impact of residual eccentricity close to merger into spin-aligned effective-one-body merger-ringdown models for binary black holes.

Figures (19)

• Figure 1: Example of an eccentric trajectory of a small mass $\mu = 10^{-3}$. In the top panel we show the trajectory in the equatorial plane of the Kerr BH (black disk), evolved for five radial cycles before the LSO and then plunging and merging in the BH. The trajectory is characterized by a spin $a = 0$ of the central BH, an eccentricity $e_{\rm LSO} = 0.5$ and a relativistic anomaly $\xi_{\rm LSO} = \pi$ at the LSO. In the bottom panel we show the evolution along the trajectory of $\Delta \mathcal{E}_{\rm UCO}$, defined in Eq. \ref{['eq.: DeltaEUCO def']}, with respect to the radial coordinate $r$. During the inspiral phase $\Delta \mathcal{E}_{\rm UCO}<0$, consistent with the bound motion; the LSO crossing occurs when $\Delta \mathcal{E}_{\rm UCO}=0$ (dashed horizontal line), as expressed in Eqs. \ref{['eq:LSO E and L']}. The vertical blue band (around $r\approx 4.5$) marks the interval where the effects of the eccentric corrections to the RR force in Eqs. \ref{['eq:full RR force']} are smoothly switched off, while the vertical red band (around $r\approx 3.2$) indicates the region where the full RR force is smoothly suppressed.
• Figure 2: Comparison of the $h_{22}$ Teukolsky waveform mode with three different EOB MR model prescriptions. The top panel shows the real part of $h_{22}$: the Teukolsky waveform is shown in black, the SEOBNRv5HM QC MR model attached using QC IVs at $t_{\rm peak}^{22}$ in red, the same QC MR model attached using the eccentric IVs measured from the numerical waveform in green, and the SEOB-TMLE MR model developed in this work in blue. The system considered has $a = 0.30$ , $e_{\rm LSO} = 0.50$ and $\xi_{\rm LSO} = \pi$ and all waveforms are aligned at the peak of the mode, $t = t_{\rm peak}^{22}$. The bottom panels show the fractional amplitude difference $\Delta |h_{22}|/|h^{\rm Teuk}_{22}|$ and the phase difference $\Delta\phi_{22}$. The QC MR model exhibits clear discrepancies in the post-merger regime, which are only partially reduced when eccentric IVs are employed, while the SEOB-TMLE MR model provides improved agreement in both amplitude and phase.
• Figure 3: Effect of the value of eccentricity $e_{\rm LSO}$ on the MR features of the $h_{22}$ mode. The waveform amplitude $|h_{22}|$ (top panels) and the instantaneous GW frequency $\omega_{22}$ (bottom panels) are shown as functions of time relative to the time of the peak of the orbital frequency, i.e. as functions of $t - t_{\Omega_{\rm peak}}$. Results are shown for three values of the spin: $a=-0.70$ (left column), $a=0.00$ (central column), and $a=0.70$ (right column). All waveforms are characterized by a relativistic anomaly $\xi_{\rm LSO}=\pi$ and span eccentricities in the range $e_{\rm LSO}\in[0.0,0.9]$. Increasing eccentricity leads to an enhancement of the value of the peak amplitude near merger and to an increasing time separation between the time of the amplitude peak $t_{\rm peak}^{22}$ and $t_{\Omega_{\rm peak}}$. In the ringdown regime, $\omega_{22}$ settles around the real part of the dominant QNM frequency $\sigma_{22}$, whose value $\sigma^{\rm R}_{22}$ is indicated in the bottom panels by dashed green horizontal lines. The oscillations observed in $\omega_{22}$ around $\sigma^{\rm R}_{22}$ have an amplitude that depends on the spin and reflect the QNM mixing introduced in Sec. \ref{['sec.:Anatomy of the ringdown']}. For a fixed spin, the frequencies corresponding to different values of $e_{\rm LSO}$ largely overlap during the QNM-dominated interval, indicating that eccentricity has a weak impact on the relative excitation of the QNMs. At late times, particularly for $e_{\rm LSO}\gtrsim0.7$, enhanced oscillations in $\omega_{22}$ are observed (especially for the $a=0.00$ and $a=0.70$ cases) and are associated with earlier tail excitation and its interference with QNM contributions, as pointed out in the main text of Sec. \ref{['sec.:impact of eccentricity on the merger-ringdown']}.
• Figure 4: Dependence of the QNMs excitation on $e_{\rm LSO}$, for the case $a=-0.70$ and $\xi_{\rm LSO}=\pi$. The top panel shows the absolute values of the excitation amplitudes $|A_{\mathfrak{l} m n p}|$ introduced in Eq. \ref{['eq: QNM spheroidal decomp']} for selected fundamental prograde and retrograde QNMs with $m = 2$, plotted as functions of $e_{\rm LSO}$. In the bottom panel we plot the ratios of the coefficients $|A_{\mathfrak{l} m n p} \, \mu_{m 2 \mathfrak{l} n }(a \sigma_{\mathfrak{l} m n p})|/|A_{2201}\mu_{2 2 2 0}(a \sigma_{2 2 0 1})|$, which contain the spheroidal-spherical harmonic mixing factors $\mu_{m 2 \mathfrak{l} n }(a \sigma_{\mathfrak{l} m n p})$ for the $h_{22}$ spherical mode. While increasing eccentricity leads to a systematic enhancement of the absolute excitation amplitudes $|A_{\mathfrak{l} m n p}|$ of all considered modes, their relative excitation remains nearly unchanged over the full range of $e_{\rm LSO}$. This indicates that eccentricity primarily affects the overall strength of QNM excitation, while leaving the hierarchy and mixing structure of the dominant ringdown modes largely unaltered. The amplitudes showed in this figure are referred at the light-ring crossing time of the TM.
• Figure 5: Effect of the value of the relativistic anomaly $\xi_{\rm LSO}$, on the MR features of the $h_{22}$ mode. Similarly to Fig. \ref{['fig:effects of eccentricity']}, we plot the waveform amplitude $|h_{22}|$ (top panels) and the instantaneous GW frequency $\omega_{22}$ (bottom panels) as functions of time relative to the peak of the orbital frequency, $t-t_{\Omega_{\rm peak}}$. We consider three values of the spin, with $a=-0.70$ shown in the left column, $a=0.00$ in the central column, and $a=0.70$ in the right column. For each spin configuration, we fix three different values of the eccentricity $e_{\rm LSO}$: solid lines correspond to $e_{\rm LSO}=0.10$, dashed lines to $e_{\rm LSO}=0.50$, and dotted lines to $e_{\rm LSO}=0.80$. The relativistic anomaly spans the interval $\xi_{\rm LSO}\in[0,2\pi)$ and is represented by using different shades of red, with the lightest tone corresponding to $\xi_{\rm LSO}=0.000$ and the darkest tone to $\xi_{\rm LSO}=5.760$. For the cases $a=-0.70$ and $a=0.00$, variations in $\xi_{\rm LSO}$ do not produce visible changes in either the amplitude or frequency features of the MR signal within the time interval $(t-t_{\Omega_{\rm peak}})\in[-10,100]$. For $a=0.70$, this behavior persists at low eccentricity ($e_{\rm LSO}=0.10$), while for $e_{\rm LSO}\ge0.50$ the merger features show a dependence on $\xi_{\rm LSO}$, reflected in shifts in the timing and in differences in the height of the amplitude peak. The strongest deviations occur for values of $\xi_{\rm LSO} \simeq 0$, where an extended quasi-circularization of the trajectories at the UCO leads to a visible flattening of the waveform prior to the plunge regime. In contrast, the post-merger amplitude and frequency remain unaffected by $\xi_{\rm LSO}$. In the ringdown regime, $\omega_{22}$ settles around the real part of the dominant QNM frequency $\sigma_{22}$, whose value $\sigma^{\rm R}_{22}$ is indicated in the bottom panels by dashed green horizontal lines. The oscillations observed in $\omega_{22}$ around $\sigma^{\rm R}_{22}$ have an amplitude that depends on the spin and reflect the QNM mixing introduced in Sec. \ref{['sec.:Anatomy of the ringdown']}.