A complete measurement of a black-hole recoil through higher-order gravitational-wave modes

Abstract

General relativity predicts that gravitational waves (GWs) carry linear momentum. Consequently, the remnant black hole of a black-hole merger can inherit a recoil velocity or ``kick'' of crucial implications in, e.g., black-hole formation scenarios. While the kick magnitude is determined by the mass ratio and spins of the source, estimating its direction requires a measurement of the \textit{two orientation angles} of the source. While the orbital inclination angle is commonly reported in GW observations, the scientific potential of the azimuthal one has not been exploited to date. We show how the presence of more than one GW emission mode allows one to constrain this angle and, consequently, the kick direction of a real GW event. We analyse the GW190412 signal, which contains higher-order modes, with a numerical-relativity surrogate waveform model for black-hole mergers. We rule out kick magnitudes below the typical escape velocity of dense globular clusters $v_{\text{esc}}\approx 50$\,km/s with a Bayes Factor of $\simeq 21$ (or $\simeq 95\%$ probability). The kick forms angles $θ_{KL}^{-100M}=32^{+35}_{-14}\,°$ with the orbital angular momentum defined at a reference time $t_{\rm ref}=-100\,M$ before merger (with $M$ denoting the system mass in geometric units), $θ_{KN}=44^{+19}_{-17}\,°$ with the line-of-sight. The projections of the kick and line-of-sight onto the orbital plane form an angle $φ_{KN}^{-100M}=69^{+33}_{-38}\,°$. All quantities are quoted at a $90\%$ credible level. Finally, by analyzing numerically simulated signals, we show that recoils can be estimated in an unbiased way using the NRSur7dq4 waveform model. We briefly discuss the potential application of this type of measurement for multi-messenger observations of black-hole mergers occurring in Active Galactic Nuclei.

Figures (8)

• Figure 1: Sketch of our black-hole merger reference frame The polar and azimuthal angles $(\iota,\,\phi_N)$ characterise the orientation of the orbital plane of a black-hole merger or, conversely, the direction of the line-of-sight $\vu*{N}$ on its sky. The vector $\vu*K$ represents the final black-hole recoil (or kick). Its direction on the binary's sky is characterised by $\theta_{KL}$ and $\phi_{K}$. Finally, we characterise its direction with respect to the line-of-sight by the angles $\theta_{KN}$ and $\phi_{KN}$.
• Figure 2: GW190412 as observed on Earth and 180 deg away: impact of higher-order modes. The left panel shows the whitened LIGO Livingston data around GW190412 together with the corresponding top-100 highest likelihood (best fit) waveforms in blue. In green, we also show the signals emitted by the source in the direction opposite to the line-of-sight, which clearly differ. The mid-bottom panel zooms into the late-inspiral and merger region to highlight morphological differences. Finally, the right-bottom panel shows the corresponding Fourier transforms. The top central and right panels show the same as the bottom ones, but restricting the waveforms to only the dominant quadrupole $(2,\pm 2)$ modes. This removes the information about the azimuthal angle, making the two sets of waveforms indistinguishable.
• Figure 3: Azimuthal angle around GW190412: impact of higher-order modes and precession. The left panel shows the posterior distributions of the azimuthal location of Earth $\phi_N^{-100\,M}$ around GW190412, defined as the angle between the projection of the line-of-sight onto the orbital plane and the line joining the two BHs at a time $t_{\rm ref}=-100\,M$ before merger. We show this for analyses including higher modes and precession, ignoring higher modes and ignoring both effects. The filled histograms in the central and rightmost panels show the same quantity, but also computed at times $t_{\rm ref}=-500\,M$ and $-1000\,M$. The empty histograms, instead, show the angle formed by the projections of the kick and the line-of-sight on the orbital plane $\phi_{KN}^{t_{\rm ref}}$ (solid) and the plane normal to the total angular momentum $\vb*{J}$ (dashed) $^{J}\phi_{KN}^{t_{\rm ref}}$. In the central panel, we ignore orbital precession in the analysis while in the right panel we include it.
• Figure 4: Magnitude and direction of the GW190412 recoil. The side panels show the one-dimensional posterior distribution for the magnitude of the kick of GW190412 (in terms of its base-10 logarithm), together with those for the angles it forms with the line-of-sight $\theta_{KN}$, the orbital plane $\theta_{KL}^{-100M}$, and the projection of the former onto the latter $\phi^{-100M}_{KN}$. The orbital plane is defined at a reference time $t_{\rm ref}=-100\,M$ before merger. The inner panels show the corresponding 2-dimensional $90\%$ credible regions. We show in grey the corresponding priors for the case where precession is included.
• Figure 5: Discarding small recoil magnitudes. The top panel shows the prior and posterior distributions for the kick magnitude $K$ of GW190412, zooming in the $K<120$ km/s region. The bottom panel shows the Bayes Factor ${\cal{B}}^{+}_{-}(K_0)$, between template models restricted to sources above and below a given threshold $K_0$. The red line shows the result obtained from the posterior and prior shown in the top panel. The blue line and contours denote the median and $90\%$ credible bounds obtained through the generation of 500 random re-samplings. Finally, the two horizontal lines denote Bayes factor values of 1 (equal preference) and 21.