Asymptotically matched quasi-circular inspiral and transition-to-plunge in the small mass ratio expansion

In the small mass ratio expansion and on equatorial plane, two-body problem for point particles in general relativity admits a quasi-circular inspiral motion followed by transition-to-plunge motion. We first derive equations governing Kerr background at adiabatic, post-adiabatic post-post-adiabatic orders slow-timescale terms of self-force we highlight structure higher subleading orders. parallel to any order, demonstrate that they are governed sourced linearized Painlevé transcendental kind. The ten perturbative do not require further developments theory, as determined second-order self-force. propose scheme matches with all overlapping region exterior last stable orbit where both expansions valid. explicitly verify validity matching conditions large set coefficients involved, one hand, adiabatic or and, other leading, This result is instrumental deriving gravitational waveforms within formalism beyond innermost circular orbit.