The typical mass ratio and typical final spin in supermassive black hole mergers

We prove that merging supermassive black holes (SMBHs) typically have neither equal masses, nor is their mass ratio too extreme. The majority of such mergers fall into the mass ratio range of 1:30 to 1:3, implying a spin flip during the inspiral. We also present a simple expression for the final spin $\chi_{f}$ of the emerging SMBH, as function of the mass ratio, initial spin magnitudes, and orientation of the spins with respect to the orbital plane and each other. This formula approximates well more cumbersome expressions obtained from the fit with numerical simulations. By integrating over all equally likely orientations for precessing mergers we determine a lower approximant to the final spin distribution as function of the mass ratio alone. By folding this with the derived mass ratio dependent merger rate we derive a lower bound to the typical final spin value after mergers. We repeat the procedure deriving an upper bound for the typical spin in the case when the spins are aligned to the orbital angular momentum, such that there is no precession in the system. Both slopes of $\chi_{f}$ as function of the initial spins being smaller than one lead to two attractors at $\chi_{f}^{prec}=0.2$ and $\chi_{f}^{align}=0.45$, respectively. Real mergers, biased toward partial alignment by interactions with the environment (accretion, host galaxy, etc.) would generate a typical final spin lying between these two limiting values. These are the typical values of the spin after the merger, starting from which the spin can built up by further gaseous accretion.