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

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 due to non-vacuum environmental interactions, as well as post-Newtonian (PN) spin-orbit couplings. 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. In this paper 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 which concentrated on the antialigned portion of this sequence, to arrive at a consistent maximum recoil of $448\pm 5\mathrm{km}/\mathrm{s}$ for antialigned models as well as to a phenomenological expression for the recoil velocity as a function of spin ratio. Quite surprisingly, 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, without which no systematic behavior can be found, 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, by decomposing the recoil computation into coupled modes, 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 $\ell =3$. 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.

Figure 1

The $L_{\infty }$ norm of the Einstein tensor Eq. (19) 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 $\mathcal{O}(300)$ 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. (a1, a2, a3). 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 quasinormal ringing and shows that it is well-captured at all resolutions.

Figure 4

Amplitude of $r_{\mathrm{E},\mathrm{sch}}|{\Psi }_4|$ for extraction spheres at $r_{\mathrm{E}}=30M$, $40M$, $50M$, and $60M$, 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 $r_{\mathrm{E}}=50M$ 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.

Figure 6

Left panel: Evolution in velocity space of the recoil-velocity vector. Very little variation is recorded before the radiation reaches the observer at $r_{\mathrm{E}}=50M$ (dotted lines in the two insets). The absence of the proper linear momentum in the initial data triggers a rapid and an almost straight-line motion (dashed line) of the center of the spiral away from the origin of coordinates during the initial stages of the evolution. After this transient motion, the evolution is slower, with the spiral progressively opening up (solid line). The vector to the center of the spiral corresponds to the initial linear momentum of the spacetime and is used as integration constant for Eqs. (28, 35). The final part of the evolution is characterized by a change in the spiral pattern (long-dashed line) as a result of the interaction of different modes in the ringdown of the final black hole. Note that the figure has been rotated clockwise of about 30° to allow for the two insets. Right panel: Initial behavior of the recoil velocity (upper graph) and of the waveform ($Q_{22}^{+}$) for model $r0$ (lower graph). This figure should be compared with the initial vector evolution of the recoil velocity shown in the left panel where the same types of lines have been used for the different stages of the evolution.

Figure 7

Left panel: The same as in the left panel of Fig. 6 but for system $r7$. Shown in the inset is the sudden reorientation of the recoil velocity vector during ringdown and corresponding to a new spiral with different aperture (long-dashed line). Although more pronounced in $r7$, the appearance of this hook at ringdown is seen all the members of the sequence. Right panel: The same as in the left panel of Fig. 6 but for system $r7$. The upper graph concentrates on the final stages of the evolution in of the recoil velocity and on the appearance of a second peak during ringdown (long-dashed curve). The lower graph shows the same but in terms of the $Q_{22}^{+}$ waveform. A discussion of these final stages of the evolution is made in Sec. 4c.

Figure 8

Left panel: Recoil velocity as a function of the spin asymmetry parameter $a_1/a_2$ for the models listed in Table . Indicated with a continuous lines are the results obtained via ${\Psi }_4$, while a dashed line is used for the gauge-invariant quantities $Q_{\ell m}^{+,\times }$. Right panel: Final recoil velocity calculated with both the use ${\Psi }_4$ (empty circles) and the gauge-invariant quantities (stars). Shown in the inset is the incorrect scaling obtained when the correction for the integration constant is not made.

Figure 9

Upper panel: Comparison of the computed data for the recoil velocity (open circles) with the least-squares fits using either a linear (dotted line) or a quadratic dependence (dashed line). Lower panel: Point-wise residuals computed with the linear (dotted line) or a quadratic fit (dashed line).

Figure 10

The total kick calculated via Eq. (35) up to $\ell =7$ is compared to the contributions of individual terms $q_1$ and $q_2$, as well as the sum of term excluding these. In the case of the $r0$ system (left panel) the spins are antialigned and the $q_2$ term is dominant and the $q_1$ term does not provide a significant contribution. In the case of the $r7$ system (right panel), on the other hand, the spins are essentially aligned and the while the $q_2$ term is still dominant, the $q_1$ term also makes a significant contribution.

Figure 11

Left panel: Dependence on the spin ratio of the initial total angular momentum $J_{\mathrm{ini}}$ [as computed from Eq. (47)], of the radiated angular momentum $J_{\mathrm{rad}}$ [as computed through the gauge-invariant waveforms], and of the final spin of the black hole $J_{\mathrm{fin}}$. All quantities show a linear behavior, whose coefficient are collected in Table . Right panel: Relative error $\Delta J/J_{\mathrm{ini}}$ in the conservation of the angular momentum [cf., Eq. (54)]. Different curves refer to whether the final spin of the black hole is computed using the isolated/dynamical horizon formalism (triangles) or the distortion of the apparent horizon (squares). In both cases the error is of about 1% at most for simulations at the medium resolution.

Figure 12

Left panel: Dependence on the spin ratio of the ADM mass $M_{\mathrm{ADM}}$, of the scaled radiated energy $M_{\mathrm{rad}}$ [as computed through the gauge-invariant waveforms and scaled by a factor of 10 to make it visible], and of the final mass of the black hole $M_{\mathrm{fin}}$. All quantities show linear behaviors, whose coefficients are collected in Table . Right panel: Relative error $\Delta M/M_{\mathrm{ini}}$ in the conservation of the energy [cf., Eq. (56)]. Note that the error is of about 0.5% at most for simulations at the medium resolution.

Figure 13

Left panel: Evidence that the conditions for the Peeling theorem are met also for ${\Psi }_3$, which scales as $r^{-2}$ when extracted at isotropic radii $r_{\mathrm{E}}=30M$, $40M$, $50M$, and $60M$. This figure should be compared to the corresponding Fig. 4. Right panel: The same as the left panel but for the gauge-invariant quantity $Q_{22}^{+}$, which is shown to be constant when extracted at isotropic radii $r_{\mathrm{E}}=30M$, $40M$, $50M$, and $60M$.

Figure 14

Comparison of the two polarization amplitudes $h_{+}$ (upper graph) and $h_{\times }$ (lower graph) as computed with ${\Psi }_4$ (continuous black line) or with the gauge-invariant quantities $Q_{\ell m}^{+}$ (dashed red line). Note the two polarizations are computed using the lowest (and dominant) multipole $\ell =2$, $m=2$ and are extracted at $r_{\mathrm{E}}=50M$.

Figure 15

Left panel: Coordinate trajectories for one of the black holes for the $r0$ compared with similar models where the initial linear momenta have been changed by $\pm 3\%$ in order to modify the eccentricity of the inspiral. Right panel: Recoil velocity for the $r0$ case is compared with similar models for which the initial eccentricity has been increased by adding and subtracting 3% of the initial linear momentum of the black holes relative to the $r0$ values. The effect of increased eccentricity in the final merger is to increase the size of the kick, by about 4% in both cases.