The most complicated binary black hole source is an eccentric binary where both spins and the orbital plane are precessing. These sources become more important at lower frequencies, so modeling them is paramount to the LISA mission and for the prospect of multi-band gravitational wave observations. Complicated precession dynamics make it challenging to analytically model these orbits. I will present that the eccentric, fully precessing black hole binary problem is integrable in post-Newtonian theory (PN). Analogously, in the small mass-ratio (SMR) limit, the orbit of a small body with a small spin around a large spinning black hole is also integrable. This integrability lets us construct action-angle variables and frequencies, which make the evolution simple for arbitrarily long times, without resorting to averaging.

Professor Leo Stein, University of Mississippi at Oxford