Spin Diagrams for Equal-Mass Black-Hole Binaries with Aligned Spins

TL;DR

The paper develops spin diagrams for equal-mass binary black holes with spins aligned to the orbital angular momentum to predict the recoil velocity $|v_{\rm kick}|$ and final spin $a_{\rm fin}$ as functions of the initial spins $a_1$ and $a_2$. Using CCATIE numerical-relativity simulations across five spin-sequences, the authors fit compact analytic expressions: $|v_{\rm kick}|=|c_1(a_1-a_2)+c_2(a_1^2-a_2^2)|$ with $c_1=-220.97\pm0.78$, $c_2=45.52\pm2.99$, and $a_{\rm fin}=p_0+p_1(a_1+a_2)+p_2(a_1+a_2)^2$ with $p_0=0.6883\pm0.0003$, $p_1=0.1530\pm0.0004$, $p_2=-0.0088\pm0.0005$. The results indicate a maximum recoil of $|v_{\rm kick}|\approx 442$ km s$^{-1}$ for $a_1=-a_2=1$, and a final-spin range $a_{\rm fin}\in[0.347,0.959]$, with symmetry reducing the parameter space. These spin-diagrams enable quick interpretation and may inform astrophysical scenarios such as SMBH retention and QSO activity post-merger, while acknowledging the need for broader coverage including unequal masses and generic-spin configurations.

Abstract

Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest as they may be the preferred end-state of the inspiral of generic supermassive binary black-hole systems. In view of this, we have computed the inspiral and merger of a large set of binary systems of equal-mass black holes with spins aligned with the orbital angular momentum but otherwise arbitrary. By least-square fitting the results of these simulations we have constructed two "spin diagrams" which provide straightforward information about the recoil velocity |v_kick| and the final black-hole spin a_fin in terms of the dimensionless spins a_1 and a_2 of the two initial black holes. Overall they suggest a maximum recoil velocity of |v_kick|=441.94 km/s, and minimum and maximum final spins a_fin=0.3471 and a_fin=0.9591, respectively.

Figures (3)

• Figure 1: Position in the $(a_1,\,a_2)$ space of the five sequences $r,\, ra,\, s,\, t$, and $u$ for which the inspiral and merger has been computed.
• Figure 2: Contour plots of $|v_{\rm kick}|$ as a function of the spin parameters $a_1$ and $a_2$. The diagram has been computed using expressions (\ref{['vk_3']}) and (\ref{['vk_fit_2']}).
• Figure 3: Contour plots of $a_{\rm fin}$ as a function of the spin parameters $a_1$ and $a_2$. The diagram has been computed using expressions (\ref{['af_2']}) and (\ref{['af_fit_2']}).