Statistical studies of spinning black-hole binaries

Abstract

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 ${\stackrel{\ensuremath{\rightarrow}}{S}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{S}}_{2}$ over the sphere and magnitudes $|{\stackrel{\ensuremath{\rightarrow}}{S}}_{1}/{m}_{1}^{2}|=|{\stackrel{\ensuremath{\rightarrow}}{S}}_{2}/{m}_{2}^{2}|=0.97$ for different mass ratios, where ${\stackrel{\ensuremath{\rightarrow}}{S}}_{i}$ 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, $\stackrel{\ensuremath{\rightarrow}}{L}$. We perform ${16}^{4}=65\text{ }536$ simulations for six mass ratios between $q=1$ and $q=1/16$ and compute the distribution of the angles $\stackrel{^}{\stackrel{\ensuremath{\rightarrow}}{L}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}}$ and $\stackrel{^}{\stackrel{\ensuremath{\rightarrow}}{L}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\stackrel{\ensuremath{\rightarrow}}{S}}$, 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 $P(v)\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}v/2500\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1})$ with mean velocity $⟨v⟩=630\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ and standard deviation $\sqrt{⟨{v}^{2}⟩\ensuremath{-}⟨v{⟩}^{2}}=534\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$, leading to a 23% probability of recoils larger than $1000\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$, 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\ifmmode^\circ\else\textdegree\fi{} 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.