Recoil velocities from equal-mass binary black-hole mergers: a systematic investigation of spin-orbit aligned configurations

Abstract

Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest, as studies indicate that this configuration is preferred in nature. If the spins of the two bodies differ, there can be a prominent beaming of the gravitational radiation during the late plunge, causing a recoil of the final merged black hole. We perform an accurate and systematic study of recoil velocities from a sequence of equal-mass black holes whose spins are aligned with the orbital angular momentum, and whose individual spins range from a = +0.584 to -0.584. In this way we extend and refine the results of a previous study and arrive at a consistent maximum recoil of 448 +- 5 km/s for anti-aligned models as well as to a phenomenological expression for the recoil velocity as a function of spin ratio. This relation highlights a nonlinear behavior, not predicted by the PN estimates, and can be readily employed in astrophysical studies on the evolution of binary black holes in massive galaxies. An essential result of our analysis is the identification of different stages in the waveform, including a transient due to lack of an initial linear momentum in the initial data. Furthermore we are able to identify a pair of terms which are largely responsible for the kick, indicating that an accurate computation can be obtained from modes up to l=3. Finally, we provide accurate measures of the radiated energy and angular momentum, finding these to increase linearly with the spin ratio, and derive simple expressions for the final spin and the radiated angular momentum which can be easily implemented in N-body simulations of compact stellar systems. Our code is calibrated with strict convergence tests and we verify the correctness of our measurements by using multiple independent methods whenever possible.

Figures (15)

• Figure 1: The $L_\infty$ norm of the Einstein tensor Eq. \ref{['eq:EinsteinNorm']} as a function of time. During the periods of strong dynamics (i.e., when the time derivatives of the evolution variables are large) the convergence order is dominated by the accuracy of the time-interpolation algorithm used at mesh refinement boundaries, thus yielding third-order accuracy. At the times when these time-derivatives are small, the fourth-order finite-differencing algorithm becomes the dominant source of the error. Note that the very large violations (of ${\cal O}(300)$ at the medium resolution) are confined to a single grid point on the trailing edge of the apparent horizon and are produced by the very steep gradients in the shift. As discussed later, this does not affect the fourth-order convergence of the waveforms. At the time of the merger a common apparent horizon forms and its excision from the calculation of the $L_\infty$ norm is responsible for the drop in the violation.
• Figure 2: Convergence of the fiducial waveform $Q^+_{22}$ for the binary system $r0$ before and after the time-shift defined in Eqs. \ref{['eq:timeshiftingdef']}--\ref{['eq:timeshiftingval']}. In the upper graph we show the difference between $Q^{+}_{22}$ when computed at different resolutions, scaled for fourth-order convergence and using raw data (i.e., without time-shifting). The overlap between the curves is rather poor indicating an over-convergence (i.e., the truncation error appears to be smaller than expected). In the lower panel we show the same data but after time-shifting. The very good overlap of the scaled curves on the indicates that the time-shifting is essential for obtaining properly scaling differences between runs of various resolutions.
• Figure 3: Accuracy of the fiducial waveform $Q^+_{22}$ for the binary system $r0$. In the upper graph we show the waveforms at the three different resolutions: very-high (continuous line), high (dashed line), medium (dotted line). The accuracy is very good already with the lowest resolution and the curves cannot be distinguished. The lower panels show magnifications of some relevant portions of the waveform, with the lower-left panel concentrating on the initial transient radiation produced by the truncation error. The lower-right panel, on the other hand, refers to the quasi-normal ringing and shows that it is well-captured at all resolutions.
• Figure 4: Amplitude of $r_{_{\rm E, sch}} |\Psi_4|$ for extraction spheres at $r_{_{\rm E}}=30\,M$, $40\,M$, $50\,M$ and $60\,M$, demonstrating that $\Psi_4$ does indeed fall off as required by the peeling property. There is a slight decrease in amplitude with larger radius, suggesting that dissipative effects may become important at larger radii. Results in this paper use waveforms from the $r_{_{\rm E}}=50\,M$ extraction sphere, unless indicated otherwise.
• Figure 5: The recoil velocity of the binary $r0$ is compared to those of the same system but with either a larger or a smaller initial separation (i.e., $r0l$ and $r0s$, respectively). Note the same recoil velocity is obtained when the integration constant is properly taken into account, while an error as large as $\sim 13\%$ is made otherwise.