Eccentricity and Inclination Excitation of Stellar Orbits around merging Black-Hole Binaries

Identifying the formation channels of merging compact-object binaries remains a fundamental challenge in gravitational-wave (GW) astrophysics. Here, we formulate a generalized framework incorporating both third-body perturbations and post-Newtonian effects to investigate the resonance-driven excitation of eccentricity and/or inclination for stellar orbits around merging black-hole binaries (BHBs). We show that at the quadrupole level, inclination excitation is driven by adiabatic capture into a quadrupole-order resonance, with subsequent resonance escape governed by the conservation of phase-space area. Furthermore, in the coplanar octupole regime, an apsidal precession resonance triggers robust eccentricity growth, where the capture probability depends on the initial conditions. For spatial octupole configurations, we reveal a distinct two-stage evolutionary pathway: the quadrupole-order resonance first excites the orbital inclination, followed by subsequent eccentricity growth driven by the inverse von Zeipel-Lidov-Kozai (ZLK) resonance. Our results indicate that these resonance-driven dynamics could imprint observable kinematic signatures on surrounding stellar populations, thereby offering a novel indirect probe for merging BHBs.