Resonant excitation of eccentricity in spherical extreme-mass-ratio inspirals

Gravitational radiation reaction has been one of the fundamental issues in general relativity. Over a span of decades, this process has been analyzed in the adiabatic limit in order to comprehend how it drives extreme-mass-ratio binaries that are prime targets for spaceborne detectors. It has been shown that spherical orbits around Schwarzschild and Kerr black holes remain spherical (zero eccentricity) under the influence of gravitational radiation reaction. Here, we show that spherical orbits in non-Kerr black holes that still preserve most of the good qualities and symmetries of Kerr spacetime can access certain resonances in such a way that an initially spherical inspiral acquires nonzero eccentricity and becomes nonspherical. Therefore, the crossing of resonances under radiation reaction interrupts and even inverts, up to some small radius close to plunge, the process of circularization of orbits. The strength of resonant excitation of eccentricity depends on the initial position and inclination of the integrable extreme-mass-ratio system, as well as the integrability-breaking parameter introduced in the background spacetime that amplifies further the excitation. We find that the harmonics of gravitational waves emitted from these inspirals undergo a frequency modulation as the orbit ``metamorphoses'' from spherical to nonspherical, due to the effect of resonant eccentricity excitation. The gain that low-amplitude harmonics experience in these oligochromatic extreme-mass-ratio inspirals, due to resonances, may be detectable with future spaceborne detectors and serves as an indicator of non-Kerrness of the background spacetime.