Anatomy of the binary black hole recoil: A multipolar analysis

TL;DR

The study develops a robust multipolar framework to dissect gravitational recoil in binary black hole mergers, integrating NR results with Thorne's radiative-moment formalism. By showing that multipoles up to $l=4$ suffice and that three dominant mode-pairs govern the kick, the authors connect the inspiral dynamics to a Kerr QNM-based ringdown and demonstrate an effective Newtonian approach that captures the amplitude and phase behavior across the transition. The work explains the anti-kick through mode-phase interactions and clarifies why planar versus non-planar spins yield drastically different kicks, including the large kicks seen in special configurations. These insights enable more accurate analytic modeling (PN/EOB) of recoil, with practical implications for SMBH retention in galaxies and gravitational-wave source characterization for facilities like LISA.

Abstract

We present a multipolar analysis of the gravitational recoil computed in recent numerical simulations of binary black hole (BH) coalescence, for both unequal masses and non-zero, non-precessing spins. We show that multipole moments up to and including l=4 are sufficient to accurately reproduce the final recoil velocity (within ~2%) and that only a few dominant modes contribute significantly to it (within ~5%). We describe how the relative amplitudes, and more importantly, the relative phases, of these few modes control the way in which the recoil builds up throughout the inspiral, merger, and ringdown phases. We also find that the numerical results can be reproduced by an ``effective Newtonian'' formula for the multipole moments obtained by replacing the radial separation in the Newtonian formulae with an effective radius computed from the numerical data. Beyond the merger, the numerical results are reproduced by a superposition of three Kerr quasi-normal modes (QNMs). Analytic formulae, obtained by expressing the multipole moments in terms of the fundamental QNMs of a Kerr BH, are able to explain the onset and amount of ``anti-kick'' for each of the simulations. Lastly, we apply this multipolar analysis to help explain the remarkable difference between the amplitudes of planar and non-planar kicks for equal-mass spinning black holes.

Figures (20)

• Figure 1: The recoil velocity as a function of time for a binary BH system with mass ratio 1:2 and no spins. The total recoil is plotted in black, along with the contributions from different mode pairs, described below in Sec. \ref{['multipoles']}. We denote by $t_{\rm peak}$ the time at which the multipole $I^{22}$ reaches its maximum (see Section \ref{['multipoles']}).
• Figure 2: Dominant orbital angular frequency obtained from the individual radiative multipole moments, as determined by Eq. (\ref{['omega_lm']}). The different frequencies with $\ell=m$ agree closely throughout the inspiral and RD phases. The frequency with $\ell=2,m=1$ decouples from the others at earlier time and reaches a much higher plateau. The left panel refers to the NE$_{00}^{2:3}$ run and the right panel to the NE$_{00}^{1:2}$ run. We denote with $t_{\rm peak}$ the time at which $I^{22}$ reaches its maximum.
• Figure 3: Amplitudes of the dominant radiative multipole moments. On the left panel we show the modes for the NE$^{2:3}_{00}$ run, while on the right panel the modes for the NE$^{1:2}_{00}$ run. The leading-order mass quadrupole $I^{22}$ is about an order of magnitude stronger than any other mode. The oscillating behavior of the $S^{32}$ moment during RD is likely due to mode mixing with $I^{22}$. We denote with $t_{\rm peak}$ the time at which $I^{22}$ reaches its maximum.
• Figure 4: Linear momentum flux of the strongest radiative multipole moments, i.e., the ones in Eq. (\ref{['flux_approx']}). On the left panel we show the modes for the NE$^{2:3}_{00}$ run, while on the right panel the modes for the NE$^{1:2}_{00}$ run. We denote with $t_{\rm peak}$ the time at which $I^{22}$ reaches its maximum.
• Figure 5: Comparison of numerical and QNM waveforms for the NE$_{00}^{2:3}$ run. The dominant modes analyzed are ${}_{-2}C_{22}$ ( upper left), ${}_{-2}C_{33}$ ( upper right), ${}_{-2}C_{32}$ ( lower left), and ${}_{-2}C_{44}$ ( lower right). Note that the ${}_{-2}C_{32}$ waveform includes contributions from the $\ell=2,m=2$ modes as well. We denote with $t_{\rm peak}$ the time of the peak of $I^{22}$.