Statistical studies of spinning black-hole binaries

We study the statistical distributions of the spins of generic black-hole binaries during the inspiral and merger, as well as the distributions of the remnant mass, spin, and recoil velocity. For the inspiral regime, we start with a random uniform distribution of spin directions ${\overrightarrow{S}}_1$ and ${\overrightarrow{S}}_2$ over the sphere and magnitudes $|{\overrightarrow{S}}_1/m_1^2|=|{\overrightarrow{S}}_2/m_2^2|=0.97$ for different mass ratios, where ${\overrightarrow{S}}_i$ and $m_i$ are the spin-angular momentum and mass of the $i$th black hole. Starting from a fiducial initial separation of $R_i=50M$, we perform 3.5-post-Newtonian-order evolutions down to a separation of $R_f=5M$, where $M=m_1+m_2$, the total mass of the system. At this final fiducial separation, we compute the angular distribution of the spins with respect to the final orbital angular momentum, $\overrightarrow{L}$. We perform ${16}^4=65536$ simulations for six mass ratios between $q=1$ and $q=1/16$ and compute the distribution of the angles $\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }}$ and $\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{S}}$, directly related to recoil velocities and total angular momentum. We find a small but statistically significant bias of the distribution towards counteralignment of both scalar products. A post-Newtonian analysis shows that radiation-reaction-driven dissipative effects on the orbital angular momentum lead to this bias. To study the merger of black-hole binaries, we turn to full numerical techniques. In order to make use of the numerous simulations now available in the literature, we introduce empirical formulas to describe the final remnant black-hole mass, spin, and recoil velocity for merging black-hole binaries with arbitrary mass ratios and spins. Our formulas are based on the post-Newtonian scaling, to model the plunge phase, with amplitude parameters chosen by a least-squares fit of recently available fully nonlinear numerical simulations, supplemented by inspiral losses from infinity to the innermost stable circular orbit. We then evaluate those formulas for randomly chosen directions of the individual spins and magnitudes as well as the binary’s mass ratio. The number of evaluations has been chosen such that there are 10 configurations per each dimension of this parameter space, i.e. ${10}^7$. We found that the magnitude of the recoil velocity distribution decays exponentially as $P(v)\sim \exp (-v/2500\mathrm{km}{\mathrm{s}}^{-1})$ with mean velocity $⟨v⟩=630\mathrm{km}{\mathrm{s}}^{-1}$ and standard deviation $\sqrt{⟨v^2⟩-⟨v{⟩}^2}=534\mathrm{km}{\mathrm{s}}^{-1}$, leading to a 23% probability of recoils larger than $1000\mathrm{km}{\mathrm{s}}^{-1}$, and a highly peaked angular distribution along the final orbital axis. The studies of the distribution of the final black-hole spin magnitude show a universal distribution highly peaked at $S_f/m_f^2=0.73$ and a 25° misalignment with respect to the final orbital angular momentum, just prior to full merger of the holes. We also compute the statistical dependence of the magnitude of the recoil velocity with respect to the ejection angle. The spin and recoil velocity distributions are also displayed as a function of the mass ratio. Finally, we compute the effects of the observer orientation with respect to the recoil velocity vector to take into account the probabilities to measure a given redshifted (or blueshifted) radial velocity of accretion disks with respect to host galaxies.

Figure 1

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 2

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=3/4$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 3

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/2$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 4

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/4$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 5

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/8$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 6

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/16$ at $r=5M$ starting from a uniform distribution at $r=50M$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 7

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 8

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=3/4$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 9

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/2$. The data have been binned with a bin width of $\delta \mu =0.01$.

Figure 10

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/4$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 11

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/8$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 12

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/16$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 13

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/4$ at $r=5M$ starting from a uniform distribution at $r=50M$ and ${\alpha }_1={\alpha }_2=0.97/\sqrt{2}$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 14

The $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution for $q=1/4$ at $r=5M$ starting from a uniform distribution at $r=50M$ and ${\alpha }_1={\alpha }_2=0.97/2$. Here we plot the number of events in the given range of $\mu$ out of ${16}^4$ total events.

Figure 15

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/4$ and ${\alpha }_1={\alpha }_2=0.97/\sqrt{2}$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 16

The fit to the normalized $P(\mu =\widehat{\overrightarrow{L}}·\widehat{\overrightarrow{\Delta }})$ distribution at $r=5M$ for $q=1/4$ and ${\alpha }_1={\alpha }_2=0.97/2$. The data have been binned with a bin width of $\delta \mu =0.01$ and normalized to a total probability of 1.

Figure 17

The dependence of the slope in the distribution of the angle between $\overrightarrow{\Delta }$ and the orbital angular momentum, as well as the angle between $\overrightarrow{S}$ and the orbital angular momentum as a function of the mass ratio.

Figure 18

The dependence of the slope in the distribution of the angle between $\overrightarrow{\Delta }$ and the orbital angular momentum $\overrightarrow{L}$ for $q=1/4$ as a function of $|{\overrightarrow{\alpha }}_1|=|{\overrightarrow{\alpha }}_2|=\alpha$, as well as the angle between $\overrightarrow{S}$ and the orbital angular momentum as a function of $\alpha$. In all fits the constant term is taken to be zero. The data here favor a linear dependence in $\alpha$.

Figure 19

The remnant BH spin-magnitude distribution after several generations of mergers, starting from a uniform initial distribution.

Figure 20

The remnant BH spin-magnitude distribution after several generations of mergers (starting from a uniform initial distribution) for mass ratios in the ranges $0\le q\le 0.1$ (dashed curve), $0.1\le q\le 0.2$, $0.4\le q\le 0.5$, $0.6\le q\le 0.7$, and $0.9\le q\le 1.0$. The distribution becomes more sharply peaked around larger values of $\alpha$ as the mass ratio increases. Note for the smallest range, the spin is more highly peaked than for $0.1\le q\le 0.2$.

Figure 21

The remnant spin direction distribution after several generations of mergers, starting from a uniform initial distribution, for mass ratios in the ranges $0\le q\le 0.1$, $0.2\le q\le 0.3$, and $0.9\le q\le 1$. The closer to the equal masses, the more highly peaked the distribution. For mass ratios in the range $0.9\le q\le 1$, the distribution is peaked at $\theta \sim 25°$. For the smaller mass ratios, the distribution approaches $\sin $ $\theta$ (i.e. the uniform spin direction distribution).

Figure 22

The recoil velocity magnitude distribution for a uniform distribution in mass ratio and spin-magnitude distribution in Fig. 19 (with uniform spin direction). Here $⟨v⟩=630\mathrm{km}{\mathrm{s}}^{-1}$ and $\sqrt{⟨v^2⟩-⟨v{⟩}^2}=534\mathrm{km}{\mathrm{s}}^{-1}$.

Figure 23

The recoil velocity direction distribution. The angle $\theta$ is in radians and is defined as the angle between the recoil vector and the orbital angular momentum vector.

Figure 24

Velocity distribution as a function of the angle the recoil makes with the orbital angular momentum (the distribution is symmetric with respect to $\theta \to \pi -\theta$). The plot shows $P(v)$ versus $v$ for recoils with angles in the ranges (0, 10°) (top), (10°, 20°) (2nd), (20°, 30°) (3rd), (80°, 90°) (bottom). The magnitude of $P(v)$ is the total number of simulations with recoil velocity directions between the given angles and magnitude within the range $v\pm 15\mathrm{km}{\mathrm{s}}^{-1}$. The maxima occur at $v(\mathrm{km}{\mathrm{s}}^{-1})\approx 600$, 300, 200, and 100, respectively. These distributions were obtained starting from binaries with a uniform distribution in mass ratio and distribution in spin magnitude (random directions) given in Fig. 19.

Figure 25

The recoil velocity magnitude distribution for a uniform distribution in mass ratio and spin-magnitude distribution in Fig. 19 (with uniform spin direction). The plot shows the recoil velocity distribution for mass ratios in the ranges $0\le q\le 0.1$, $0.1\le q\le 0.2$, $0.3\le q\le 0.4$, and $0.9\le q\le 1$. The distributions become successively broader for larger values of $q$ (i.e. similar masses).

Figure 26

The remnant spin direction distribution after several generations of mergers, starting from a uniform initial distribution.

Figure 27

The distribution of the recoil velocity along an observers line of sight for a uniform distribution in mass ratio and spin-magnitude distribution in Fig. 19 (with uniform spin direction). Here $⟨v⟩=315\mathrm{km}{\mathrm{s}}^{-1}$ and $\sqrt{⟨v^2⟩-⟨v{⟩}^2}=358\mathrm{km}{\mathrm{s}}^{-1}$.

Figure 28

The distribution of recoil velocity magnitudes and directions (with respect to the orbital plane) assuming a distribution in mass ratios uniform in ${log}_{10}q$ in the interval $-2\le {log}_{10}q\le 0$.

Figure 29

The polar ISCO orbit in the case of $a=0.9M$, where we set $M=1$.