Spin flips and precession in black-hole-binary mergers

We use the “moving puncture” approach to perform fully nonlinear evolutions of spinning quasicircular black-hole binaries with individual spins unaligned with the orbital angular momentum. We evolve configurations with the individual spins (parallel and equal in magnitude) pointing in the orbital plane and 45° above the orbital plane. We introduce a technique to measure the spin direction and track the precession of the spin during the merger, as well as measure the spin flip in the remnant horizon. The former configuration completes 1.75 orbits before merging, with the spin precessing by 98° and the final remnant horizon spin flipped by $\sim 72°$ with respect to the component spins. The latter configuration completes 2.25 orbits, with the spins precessing by 151° and the final remnant horizon spin flipped $\sim 34°$ with respect to the component spins. These simulations show for the first time how the spins are reoriented during the final stage of black-hole-binary mergers verifying the hypothesis of the spin-flip phenomenon. We also compute the track of the holes before merger and observe a precession of the orbital plane with frequency similar to the orbital frequency and amplitude increasing with time.

Figure 1

The puncture trajectories along with spin direction (every $4M$) for the SP3 configuration for the $M/30$ resolution run. The spins are initially aligned along the $y$-axis, but rotate by $\sim 90°$ during the 1.25 last orbits and also acquire a non-negligible $z$-component. Note that the $z$-scale is $1/10$th the $x$ and $y$ scale.

Figure 2

The projection of puncture trajectories and spin for the SP3 configuration onto the $xy$ plane along with the individual apparent horizons for the $M/30$ run. The horizons and spins are shown at $t=0,20M,40M,\dots ,160M,164M$. The first common horizon (also shown) formed at $t=164.2M$. The spins are initially aligned along the $y$-axis but rotate by $\sim 90°$ during the last 1.25 orbits. The spin of the second black hole (not shown) is equal to the first.

Figure 3

The spin components and magnitude versus time for the SP3 configuration as calculated using the coordinate rotational vectors (coord) and the poles of the approximate Killing vector (IH) for the $M/30$ resolution. The calculation of the approximate Killing vector breaks down near the merger (which occurs at $t=164.2$), but the purely coordinate-based calculation continues to produce reasonable results. Note that the direction obtained from the Killing vector oscillates about the coordinate based direction. Also note the spin has just rotated by 90° in the $xy$ plane at the time of merger. The spin magnitude remains essentially constant throughout the merger phase. The magnitude of the spin calculated from the Killing vector is coordinate invariant and, unlike the spin direction, is expected to be more accurate than the coordinate-based calculation.

Figure 4

A convergence plot of the coordinate-base ${\overrightarrow{S}}_{\mathrm{coord}}$ calculation for the SP3 configuration. Note that for our choice of resolutions, third-order convergence is demonstrated by $S(M/22.5)-S(M/25)=(S(M/25)-S(M/30))C$, where $C=0.88$. The spin is initially third-order convergent, with higher order-convergence apparent at later times.

Figure 5

The $z$-component of the Killing vector based calculation of the spin rescaled by the square of the horizon mass for the three resolutions. The curves have been translated by a distance equal to the difference in merger times of the $M/22.5$ and $M/25$ runs with the merger time of the $M/30$ run. In this configuration, unlike the previously studied aligned-spin binary, the “spin-up” appears to be large. However, in this case the “spin-up” is actually the rotation of the nearly-constant spin-vector towards the $z$-axis, rather than an increase in the spin magnitude.

Figure 6

The post-Newtonian and numerical $S_x$ and $S_y$ components of the spin of the individual horizons for the SP3 configuration as functions of the $S_z$ component of the spin (which increases monotonically in time prior to the merger). Note the very reasonable agreement for most of the evolution. The plot terminates at the merger.

Figure 7

The difference in track locations between the $M/22.5$ and $M/25$ runs as well as the difference in track locations between the $M/25$ and $M/30$ runs. The latter difference has been rescaled by $C=0.88$ to demonstrate third-order convergence. The reduced order of convergence for $X$ between $t=10M$ and $t=60M$ is likely due to the coarseness of the grid.

Figure 8

The spin direction of the individual horizons every $4M$ during the spin-precession phase and the final horizon spin direction for the SP3 configuration. The arrows indicate the spin direction only, not the magnitude. Note the continuous change in the spin direction during the precession stage and the discontinuous jump (or flip) to the remnant spin direction.

Figure 9

The real and imaginary parts of the $(\ell =2,m=1)$ mode of ${\psi }_4$ for the SP3 configuration with resolution $h=M/25$ (upper panel), as well as a convergence plot of this waveform (lower panel). Note that the waveform is fourth-order convergent (as is evident by the agreement of the two differences).

Figure 10

The $z$-component of the puncture trajectory (for the puncture initially located at $x>0$) versus time for the SP3 and SP4 configurations with resolution $h=M/25$. Note that after rescaling by $\sqrt{2}$ the two trajectories agree at early times. At later times the spin-orbit coupling in the partially aligned case delays the merger; causing the two trajectories to diverge. Both curves terminate at their respective merger times.

Figure 11

The imaginary part of the $(\ell =2,m=1)$ component of ${\psi }_4$ for the SP3 and SP4 configurations extracted at $r=15M$ with a central resolution of $M/25$. Note that between $t=50$ and $t=75$ the two waveforms scale with $\sin \vartheta$ as is evident by the good agreement between the two waveform after rescaling the SP4 waveform by $\sqrt{2}$ [i.e. $1/\sin (\pi /4)$]. However, this scaling breaks down at later times ($t>85M$) due to the differences in orbital decay arising from the increased stability of the SP4 configuration. This scaling also breaks down at early times because of the nonphysical initial data radiation pulse.

Figure 12

The spin components and magnitude of the individual horizons (the two horizons have equal spins) for the SP4 configuration up to merger (with resolution $h=M/25$). Note that in this configuration the spins rotate by 135° in the $xy$ plane prior to merger. Also note that the significant $z$-direction spin-up is once again caused by a rotation further out of the $xy$ plane. Once again the Killing vector determination for the spin direction oscillates about the coordinate definition for the spin direction. The approximate Killing vector calculation begins to break down at around $t=180M$.

Figure 13

The puncture trajectories and horizon spin direction (shown every $4M$ until merger) for the SP4 configuration. Note that the $z$-scale is $1/10$th that of the $x$ and $y$ scales.

Figure 14

The $xy$ plane projection of the orbital track, apparent horizons, and spin direction for the SP4 configuration up to merger. The horizons are given at $t=0,20M,40M,\dots ,180M,196M$. The common horizon formed at $t=195.4M$. The projected spin vector decreases in magnitude at late time due to the spin rotating further out of the $xy$ plane. Note that the spin direction precesses by $135°$ in the $xy$ plane during the merger. The spin of the second black hole (not shown) is equal to the first.

Figure 15

The spin direction of the individual horizons every $4M$ during the spin-precession phase and the final horizon spin direction for the SP4 configuration. The arrows indicate the spin direction only, not the magnitude. Note the continuous change in the spin direction during the precession stage and the discontinuous jump (or flip) to the remnant spin direction.

Figure 16

The real part of the premerger $(\ell =2,m=2)$ component of ${\psi }_4$ extracted at $r=10M$. The SP4 waveform has been translated by $37M$ and multiplied by a constant phase factor prior to taking the real part. The plunge part of the waveform begins roughly at $t=168M$. Thus there are 3.5 cycles of orbital radiation prior to the plunge (but after the initial data pulse) for SP3 and 4.5 for SP4. This number of cycles match the number of orbits in Figs. 2, 14. The initial data pulse appears to mask an additional $1/2$ cycle of orbital motion.

Figure 17

The angle in degrees between the individual apparent horizon spins and the final total angular momentum for the SP3 and SP4 configuration. The SP3 angles have been translated by $-37.21°$ so that the translated SP3 and SP4 angles agree at $t=0$. The curves terminate at the point when the isolated horizon algorithm becomes inaccurate.

Figure 18

The convergence of the Hamiltonian constraint violation for the SP3 configuration along the $x$-axis at $t=76M$ when the punctures cross the $x$-axis for the second time. The Hamiltonian constraint shows third-order convergence. Points inside the domain of dependence of the boundary have been excluded. The high-frequency features near the outer boundary are due to the extreme fisheye deresolution and converge with resolution.

Figure 19

The convergence of the momentum constraint violation for the SP3 configuration along the $x$-axis at $t=76M$ when the punctures cross the $x$-axis for the second time. The momentum constraint shows third-order convergence. Points inside the domain of dependence of the boundary have been excluded. In each panel the solid (black) curve, dotted (red) curve and dashed (blue) curves are the $M/22.5$, $M/25$, and $M/30$ constraint violations, respectively. The constraints have been rescaled by $(h_l/h{)}^3$ ($h_l=M/22.5$). The high-frequency features near the outer boundary are due to the extreme fisheye deresolution and converge with resolution.

Figure 20

The convergence of the BSSN constraint violation for the SP3 configuration along the $x$-axis at $t=76M$ when the punctures cross the $x$-axis for the second time. The BSSN constraint shows third-order convergence. Points inside the domain of dependence of the boundary have been excluded. In each panel the solid (black) curve, dotted (red) curve and dashed (blue) curves are the $M/22.5$, $M/25$, and $M/30$ constraint violations, respectively. The constraints have been rescaled by $(h_l/h{)}^3$ ($h_l=M/22.5$). The high-frequency features near the outer boundary are due to the extreme fisheye deresolution and converge with resolution.

Figure 21

A linear-least squares fit of the remnant Kerr spin parameter (left $y$-axis) and radiated energy (right $y$-axis) for the merger of equal-mass equal-spin binaries with spins pointing along (or in the opposite direction to) the orbital angular momentum. The fits have the functional form $y=c_0+c_1(S/m^2)+c_2(S/m^2{)}^2$.