  The dynamics of precessing binary black holes (BBHs) in the post-Newtonian
regime has a strong timescale hierarchy: the orbital timescale is very short
compared to the spin-precession timescale which, in turn, is much shorter than
the radiation-reaction timescale on which the orbit is shrinking due to
gravitational-wave emission. We exploit this timescale hierarchy to develop a
multi-scale analysis of BBH dynamics elaborating on the analysis of Kesden et
al. (2015). We solve the spin-precession equations analytically on the
precession time and then implement a quasi-adiabatic approach to evolve these
solutions on the longer radiation-reaction time. This procedure leads to an
innovative "precession-averaged" post-Newtonian approach to studying precessing
BBHs. We use our new solutions to classify BBH spin precession into three
distinct morphologies, then investigate phase transitions between these
morphologies as BBHs inspiral. These precession-averaged post-Newtonian
inspirals can be efficiently calculated from arbitrarily large separations,
thus making progress towards bridging the gap between astrophysics and
numerical relativity.
