Black Hole Kicks as New Gravitational Wave Observables

Generic black hole binaries radiate gravitational waves anisotropically, imparting a recoil, or kick, velocity to the merger remnant. If a component of the kick along the line of sight is present, gravitational waves emitted during the final orbits and merger will be gradually Doppler shifted as the kick builds up. We develop a simple prescription to capture this effect in existing waveform models, showing that future gravitational wave experiments will be able to perform direct measurements, not only of the black hole kick velocity, but also of its accumulation profile. In particular, the eLISA space mission will measure supermassive black hole kick velocities as low as $\sim 500\mathrm{km}{\mathrm{s}}^{-1}$, which are expected to be a common outcome of black hole binary coalescence following galaxy mergers. Black hole kicks thus constitute a promising new observable in the growing field of gravitational wave astronomy.

Figure 1

GW shift due to BH kicks (artificially exaggerated to demonstrate the key features). As the kick velocity builds up during the last few orbits and merger, the emitted GWs are progressively redshifted (left) or blueshifted (right), depending on the sign of the projection of the kick velocity ${\mathbf{v}}_{\mathbf{k}}$ onto the line of sight $\widehat{\mathbf{n}}$. This is equivalent to differentially rescaling the binary’s total mass in the phase evolution from $M$ to $M(1+{\mathbf{v}}_{\mathbf{k}}·\widehat{\mathbf{n}})$. These figures have been produced by artificially imparting kicks of ${\mathbf{v}}_{\mathbf{k}}·\widehat{\mathbf{n}}=\pm 0.5c$ to nonspinning equal-mass binaries, assuming a Gaussian kick model with $\sigma =60M$ [see Eqs. (4) and (5), with ${\alpha }_n=0$ for $n\ge 1$].

Figure 2

Mismatches introduced by BH recoils. The top panel shows the mismatch $1-\mathcal{O}$ between (i) a standard waveform of equal-mass nonspinning BH binaries of total mass $M$ and (ii) a “kicked” waveform which includes the Doppler-shifting effects of a velocity profile $v(t)$. Each line corresponds to a different kick profile $v(t)$, as shown in the bottom panel. All models shown here assume ${\alpha }_n=0$ for $n\ge 2$. The $\sigma =10M$, ${\alpha }_1/{\alpha }_0=0$ model (solid line) is used in Fig. 3.

Figure 3

Detectability of BH kicks with LIGO (left) and eLISA (right). For each simulated source we compute the overlap $\mathcal{O}$ between standard and kicked waveforms, and compare it with the SNR $\rho$. Kick velocities—here encoded in the color bar—are imparted using NR fitting formulas. BH kicks are detectable for the fraction $\mathcal{F}$ of the sources above the black line, $\mathcal{O}<1-{\rho }^{-2}$. eLISA results have here been generated with the “N2A5L6” PSD of Ref. [47].