Beyond Quadrupoles: Mapping GW190412's Kick Direction with Higher-Order Gravitational Wave Modes

Gravitational waves from generic binary black hole mergers produce a net emission of linear momentum, resulting in a kick or recoil of the final remnant black hole, whose magnitude depends on the progenitor's mass ratio and spins. However, characterising the recoil direction requires measuring both the binary's inclination angle and azimuth. While the former is routinely reported in gravitational wave observations, the latter has been to date overlooked. Here, we show how the presence of non-quadrupole radiation multipoles enables us to constrain this azimuth, thus allowing us to fully constrain the remnant's kick direction. We apply our methodology to the mass-asymmetric binary black hole merger signal GW190412, analysing it with the cutting-edge waveform model NRSur7dq4. We constrain the kick angles at a reference time of -100M before the merger with a 68\% credible level: $\theta_{KL}^{-100M} = 28^\circ [+23^\circ, -11^\circ]$, signifying an upward kick; $\theta_{KN} = 37^\circ [+15^\circ, -12^\circ]$ concerning the line-of-sight; and $\phi_{KN}^{-100M} = 46^\circ [+32^\circ, -41^\circ]$ with projection onto the former. However, the kick magnitude is barely informative with marginal support for values $\lesssim 50 \, \text{km/s}$. We conclude by briefly discussing the potential application of this type of measurement for multi-messenger observations of black-hole mergers occurring in gas-rich environments.