Tracking Black Hole Kicks from Gravitational-Wave Observations

Coalescing binary black holes emit anisotropic gravitational radiation. This causes a net emission of linear momentum that produces a gradual acceleration of the source. As a result, the final remnant black hole acquires a characteristic velocity known as recoil velocity or gravitational kick. The symmetries of gravitational wave emission are reflected in the interactions of the gravitational wave modes emitted by the binary. In this Letter, we make use of the rich information encoded in the higher-order modes of the gravitational wave emission to infer the component of the kick along the line of sight (or radial kick). We do this by performing parameter inference on simulated signals given by numerical relativity waveforms for nonspinning binaries using numerical relativity templates of aligned-spin (nonprecessing) binary black holes. We find that for suitable sources, namely those with mass ratio $q\ge 2$ and total mass $M\sim 100M_{\odot }$, and for modest radial kicks of $120\mathrm{km}/\mathrm{s}$, the 90% credible intervals of our posterior probability distributions can exclude a zero kick at a signal-to-noise ratio of 15, using a single Advanced LIGO detector working at its early sensitivity. The measurement of a nonzero radial kick component would provide the first observational signature of net transport of linear momentum by gravitational waves away from their source.

Figure 1

Impact of azimuthal angle: last cycles of a GW signal emitted by a $q=3$ nonprecessing binary in the direction of its final kick (solid) with $(\iota ,\phi )=(\pi /2,{\phi }_K)$ and opposite to it (dashed) with $(\iota ,\phi )=(\pi /2,{\phi }_K+\pi )$. Strain and time are expressed in numerical relativity units, and $t_c$ coincides with the amplitude peak of the (2,2) mode. The observed final velocities are $K_r=\pm 16\mathrm{km}/\mathrm{s}$. See Sec. 2 for a detailed definition of the angles.

Figure 2

Bank of nonprecessing numerical simulations: set of simulations used as templates and injections (black circles) in this Letter, represented by their $q$ and effective spin ${\chi }_{\mathrm{eff}}=(m_1{\chi }_{1,z}+m_2{\chi }_{2,z})/(m_1+m_2)$. Here, ${\chi }_{i,z}$ denotes the projection each BH spins along the direction of the orbital angular momentum.

Figure 3

Resolvability of orientation: overlap, maximized over masses and spins, between two fiducial signals emitted in the direction of the final kick of two binaries and our NR template bank as a function of the orientation of the templates, expressed in Hammer-Aitoff coordinates. $Y_{\mathrm{HA}}=0$ denotes edge-on orientations (or the orbital plane), whereas the top and bottom of the figure denote face-on or -off orientations. $X_{\mathrm{HA}}$ is defined so that $X_{\mathrm{HA}}=0$ corresponds to $\overline{\phi }=0$. Hence, $(X_{\mathrm{HA}},Y_{\mathrm{HA}})=(0,0)$ denotes the direction of the final kick. The fiducial signals correspond to a binary with parameters consistent with GW150914 (left panel) and a $q=3$ binary with a total mass of $M=100M_{\odot }$. The stronger, higher mode content of the $q=3$ binary leads to a stronger dependence of the signal on the orientation of the binary and a rapid degradation of the overlap for orientations different than that of the source.

Figure 4

Estimations of $K_r$ for selected injections: posterior distributions for $K_r$ for several injections corresponding to a nonspinning BBH with $q=2$ $M=100M_{\odot }$ with varying orientations with respect to the observer. We consider an observed SNR $\rho =15$.

Figure 5

Uncertainty in the measurement of $K_r$: percent relative width of the 90% credible intervals, $\mathrm{\Delta }{K}_{r,90}$, of the posterior distributions for $K_r$ for signals emitted by various sources as a function of $K_r$.