Binary black hole merger gravitational waves and recoil in the large mass ratio limit

Pranesh A. Sundararajan, Gaurav Khanna, and Scott A. Hughes

Phys. Rev. D 81, 104009 – Published 5 May 2010

Abstract

Spectacular breakthroughs in numerical relativity now make it possible to compute spacetime dynamics in almost complete generality, allowing us to model the coalescence and merger of binary black holes with essentially no approximations. The primary limitation of these calculations is now computational. In particular, it is difficult to model systems with large mass ratio and large spins, since one must accurately resolve the multiple length scales that play a role in such systems. Perturbation theory can play an important role in extending the reach of computational modeling for binary systems. In this paper, we present first results of a code that allows us to model the gravitational waves generated by the inspiral, merger, and ringdown of a binary system in which one member of the binary is much more massive than the other. This allows us to accurately calibrate binary dynamics in the large mass ratio regime. We focus in this analysis on the recoil imparted to the merged remnant by these waves. We closely examine the “antikick,” an antiphase cancellation of the recoil arising from the plunge and ringdown waves, described in detail by Schnittman et al. We find that, for orbits aligned with the black hole spin, the antikick grows as a function of spin. The total recoil is smallest for prograde coalescence into a rapidly rotating black hole, and largest for retrograde coalescence. Amusingly, this completely reverses the predicted trend for kick versus spin from analyses that only include inspiral information.

Received 1 March 2010

DOI:https://doi.org/10.1103/PhysRevD.81.104009

Authors & Affiliations

Pranesh A. Sundararajan^1,*, Gaurav Khanna², and Scott A. Hughes¹

¹Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
²Department of Physics, University of Massachusetts, Dartmouth, Massachusetts 02747, USA

^*P. A. S. is currently a Desk Strategist at Morgan Stanley and Co. Inc., 1585 Broadway, New York, NY 10036, USA

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Vol. 81, Iss. 10 — 15 May 2010

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Images

Figure 1
Example quasicircular inspiral and plunge trajectory. For this calculation, the larger black hole’s spin was set to $a = 0.3 M$ , and the binary’s mass ratio is $μ / M = 10^{- 4}$ . Left panel shows $r (t)$ , the trajectory’s radius as a function of time; right panel shows the same trajectory as viewed in the equatorial plane of the larger black hole. On the left, the dotted line at $r = 4.98 M$ labels the radius of the prograde last stable orbit. Our trajectory, which starts at $r = 5.23 M$ , executes about 25 orbits before crossing this point. The inset there zooms in on the region $t ≃ 2260 M$ , showing the smaller body’s trajectory as it approaches the event horizon at $r = 1.955 M$ . On the right, the heavy circle at $r = 1.955 M$ is the hole’s event horizon; notice how the particle quickly “locks” onto the horizon after the plunge which follows its slow inspiral.Reuse & Permissions
Figure 2
Coalescence waveform computed with perturbation theory for a binary with mass ratio $μ / M = 10^{- 4}$ , and in which the larger black hole has spin $a = 0.6 M$ . We show the $+$ polarization of the waveform as viewed in the binary’s equatorial plane; the $\times$ waveform is zero from this viewing angle. The wave is normalized by $D$ , the distance from source to observer, and the origin of the time axis is arbitrary. The waveform shown in the top panel includes contributions from all modes with $| m | \leq 6$ ; the individual contributions for $m = 1$ , $m = 2$ , and $m = 3$ are shown below. Notice how the smoothly chirping inspiral waves blend naturally into the rapidly damped ringdown which terminates this waveform.Reuse & Permissions
Figure 3
Same as Fig. 2, but the large black hole has spin $a = 0$ in this case. The frequencies which describe this wave are generically lower than those shown in Fig. 2 since the transition from inspiral to plunge happens at larger radius thanks to smaller spin parameter in this binary. In addition, the final ringdown waves damp out more rapidly than in the $a = 0.6 M$ case.Reuse & Permissions
Figure 4
Same as Figs. 2, 3, but now for black hole spin $a = - 0.6 M$ (i.e., same black hole as in Fig. 2, but now for a retrograde orbit geometry). The frequencies characterizing this wave are again lower than for the two previous examples, and the ringdown waves damp even more rapidly.Reuse & Permissions
Figure 5
Summary of recoil versus time for large black hole spins $a / M \in [- 0.9, - 0.6, - 0.3, 0, 0.3, 0.6, 0.9]$ . For each track, the time origin is arbitrary, so we have shifted the data in order to cleanly display all seven recoil trends shown here. The key feature we find is the manner in which (especially for large spin, prograde coalescences) the kick builds to a large positive value, followed by an antikick that brings the total accumulated recoil down to much smaller values. The antikick is especially strong when the spin is large and the coalescence is prograde, and is essentially nonexistent for large spin retrograde coalescence.Reuse & Permissions

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