Spin flips in generic black hole binaries

We study the spin dynamics of individual black holes in a binary system. In particular we focus on the polar precession of spins and the possibility of a complete flip of spins with respect to the orbital plane. We perform a full numerical simulation that displays these characteristics. We evolve equal mass binary spinning black holes for $t=20,000M$ from an initial proper separation of $d=25M$ down to merger after 48.5 orbits. We compute the gravitational radiation from this system and compare it to 3.5 post-Newtonian generated waveforms finding close agreement. We then further use 3.5 post-Newtonian evolutions to show the extension of this spin flip-flop phenomenon to unequal mass binaries. We also provide analytic expressions to approximate the maximum flip-flop angle and frequency in terms of the binary spins and mass ratio parameters at a given orbital radius. Finally we discuss the effect this spin flip flop would have on accreting matter and other potential observational effects.

Figure 1

Initial configuration of the simulated binary.

Figure 2

Precession of the orbital plane as displayed by the distance vector $\overrightarrow{d}={\overrightarrow{x}}_1(t)-{\overrightarrow{x}}_2(t)$.

Figure 3

The conservation of the individual masses and spin magnitudes of the holes during the binary evolution.

Figure 4

BH (1,2) spin components in the coordinate frame. The top panel is BH1 and the bottom panel is BH2.

Figure 5

Change in the spin direction (in blue) of the BH1 (with the smaller spin magnitude) in the orbital frame, $\widehat{L}$ (left), and the coordinate frame $\widehat{z}$ (right). Plotted also the directions of $\overrightarrow{L}$ in red and $\overrightarrow{J}$ in green.

Figure 6

The full numerical $(\ell =2,m=2)$ waveform (above) and $(\ell =2,m=1)$ waveform (below) as extracted by observers at $r=175M$.

Figure 7

The phase difference between the full numerically generated $(\ell =2,m=2)$ waveform (NR) and the 3.5PN waveform. Differences are notably reduced when the NR waveform, extracted at $r=175M$, is extrapolated to an infinite observer location by the perturbative formulas (PE) of 1st order, accurate to $(1/r)$ terms and second order, accurate to $(1/r^2)$.

Figure 8

The amplitude difference between the full numerically generated $(\ell =2,m=2)$ waveform (NR) and the 3.5PN waveform. Differences are notably reduced when the NR waveform, extracted at $r=175M$, is extrapolated to an infinite observer location by the perturbative formulas (PE) of 1st order, $(1/r)$, and second order, $(1/r^2)$.

Figure 9

Evolution of the eccentricity versus coordinate separation of the BHB. Dots represent measurements of the eccentricity, $e_D$, and the continuous curve a fit to its decay with $e\sim r^{1.73486\pm 0.1495}$. In comparison, the theoretical prediction is $e\sim r^{1.5833}$.

Figure 10

The mass and spin of the final merged BH.

Figure 11

Flip flop of ${\overrightarrow{S}}_1$ and ${\overrightarrow{S}}_2$ as they precess with $\overrightarrow{L}$.

Figure 12

The configuration of the spins in the orbiting frame. Spin configurations ${\overrightarrow{S}}_1$ and ${\overrightarrow{S}}_2$ relative to the orbital angular momentum $\overrightarrow{L}$. Here $\overrightarrow{S}={\overrightarrow{S}}_1+{\overrightarrow{S}}_2$.

Figure 13

$q=1$ binaries, inspiraling from $r=100M$ to $r=5M$. Initial configurations having ${\varphi }_1^0=0={\varphi }_2^0$, ${\theta }_1^0=0$, and varying ${\theta }_2^0=\beta$ and $S_1/S_2$. The central blue curve follows $\cos \beta =-S_2/S_1$ and agrees with the maximum (yellow) of the flip-flop angle of ${\theta }_2$. The other two blue lines represent the 90-deg flips level.

Figure 14

Flip-flop frequency oscillation for $q=1$ binaries, inspiraling from $r=100M$ to $r=5M$. Starting from initial ${\varphi }_1^0=0={\varphi }_2^0$, ${\theta }_1^0=0$, for different choices of ${\theta }_2^0=\beta$ and $S_1/S_2$. Different surface levels are represented in units of $10^{-7}$ for the initial frequency at $r=100M$. Thin lines are the measured flip-flop frequency during evolution and thick lines their analytic approximation for quasicircular orbits, Eq. (11).

Figure 15

The probability for the $q=1$ case of having a flip greater than a certain angle. Distribution 2D lets either ${\theta }_1^0$ or ${\theta }_2^0$ be aligned or counteraligned with $\widehat{L}$ with a randomly varied cos of the other variable and spin magnitudes ratio. There are 209,450 samples that match the analytic distribution. Distribution 6D shows the probability of flip over a generic range of spins (6 parameters) for $q=1$, totaling 16 million evolutions.

Figure 16

The probability of a flip-flop angle $\mathrm{\Delta }{\theta }_{ff}\ge x$ assuming equal masses, ($q=1$), and random initial spin orientations and magnitudes of both, the primary and secondary spins. Evolutions from initial separation $r=100M$ down to $r=5M$ are performed using 3.5PN equations of motion. The dependence with the azimuthal orientation $\mathrm{\Delta }\varphi =d\varphi$ is displayed. We observe the largest flip angles probabilities for configurations with $\mathrm{\Delta }\varphi =\pi /2$ and smallest for $\mathrm{\Delta }\varphi =0$ and $\pi$.

Figure 17

The dependence of the probabilities of a flip flop larger than a given value as a function of the initial $\mathrm{\Delta }\varphi$ angle among spins in the $L$-frame. Evolutions from $r=100M$ down to $r=5M$ are performed with 3.5PN including radiation reaction. From left to right and top to bottom for $q=0.9$, 0.8, 0.7, 0.6. We observe an insensitivity on $\mathrm{\Delta }\varphi$ for mass ratios of the binary below $q=0.7$. Similar results are observed in the $J$-frame.

Figure 18

Measured maximum flip-flop angle from PN evolutions versus analytic prediction (no fitting here). Here we display the leading coefficient of Eqs. (19) and (21) for the larger BH flip-flop angle (left) and smaller BH angle (right).

Figure 19

Measured flip-flop frequency versus analytic prediction (no fitting here). Here we display the leading coefficient of Eq. (30) for the larger BH flip-flop angle (left) and smaller BH angle (right).

Figure 20

The distributions of the flip angles larger than a given (abscissa) value for a binary evolved with 3.5PN from a separation $r=100M$ to $r=5M$ for each $0.1\le q\le 0.9$ in the $L$-frame. The upper panel assumes an initial ${\varphi }_1={\varphi }_2$ while the lower panel assumes an initial ${\varphi }_1=-{\varphi }_2$ as the resonant attractors at larger separations.

Figure 21

The distributions of the flip angles for a binary at a constant separation $r=20M$ and mass ratio $q=1/2$. Spin magnitudes and angles have been chosen uniformly and the max (small, large) flip angles is evaluated. The peak at around 41° corresponds to the smaller hole spin flip. The larger hole gives an additional peak (at around 17°) towards low angles and bulks up the middle region between peaks.

Figure 22

Probability distribution of unequal mass flip flops with respect to $L$-frame on top and with respect to $J$-frame below. These results are for initial random spin distributions evolved with 3.5PN form $100M$ of initial separation down to $5M$ (approximating the merger).