Endpoint of the up-down instability in precessing binary black holes

Binary black holes in which both spins are aligned with the binary's orbital angular momentum do not precess. However, the up-down configuration, in which the spin of the heavier (lighter) black hole is aligned (anti-aligned) with the orbital angular momentum, is unstable to spin precession at small orbital separations. We first cast the spin precession problem in terms of a simple harmonic oscillator and provide a cleaner derivation of the instability onset. Surprisingly, we find that following the instability, up-down binaries do not disperse in the available parameter space but evolve toward precise endpoints. We then present an analytic scheme to locate these final configurations and confirm them with numerical integrations. Namely, unstable up-down binaries approach mergers with the two spins coaligned with each other and equally misaligned with the orbital angular momentum. Merging up-down binaries relevant to LIGO/Virgo and LISA may be detected in these endpoint configurations if the instability onset occurs prior to the sensitivity threshold of the detector. As a by-product, we obtain new generic results on binary black hole spin-orbit resonances at 2nd~post-Newtonian order. We finally apply these findings to a simple astrophysical population of binary black holes where a formation mechanism aligns the spins without preference for co- or counteralignment, as might be the case for stellar-mass black holes embedded in the accretion disk of a supermassive black hole.