Gravitational waves from inclined orbits around Kerr black holes

Abstract

Extreme mass ratio inspirals on generic inclined orbits provide a critical dynamical probe of strong field gravity, as they uniquely break the severe parameter degeneracies inherent to planar equatorial trajectories. To isolate the exact rotational signatures, we extend the periodic-orbit framework to three-dimensional inclined trajectories in the Kerr spacetime, specifically focusing on the $r$--$θ$ orbital resonances. By mapping the geodesic motion into a co-precessing spatial framework parameterized by Mino time, we compute the emitted quadrupole waveforms and evolve the local dynamics using circularity-preserving radiation reaction. It is demonstrated that frame-dragging induces a transient dynamical tilt at periastron, which directly splits the emitted radiation into an exact five-comb spectral architecture. When this resonant structure is resolved over a single coherence time, the statistical error on the dimensionless spin is reduced to $σ_a \approx 10^{-2}$, improving upon the equatorial baseline by up to two orders of magnitude. Because any bound orbit in a static, spherically symmetric spacetime generates a maximum of three frequency combs, the detection of this five-comb hierarchy establishes a rigorous, model-independent null test to strictly exclude spherical geometries.