Environmentally induced chaos: Extreme-mass-ratio systems of rotating black holes in astrophysical environments

Extreme-mass-ratio inspirals, in which a stellar-mass object orbits a supermassive black hole, are prime sources of millihertz gravitational waves for upcoming space-based detectors. While most studies assume idealized vacuum backgrounds, realistic extreme-mass-ratio binaries are embedded in astrophysical environments such as accretion disks, stellar clusters, or dark matter spikes, disks, and halos, which can significantly alter the orbital dynamics. We explore bound geodesics around general-relativistic solutions describing rotating black holes surrounded by matter halos for the first time, mapping how environmental effects interfere with the spacetime symmetries of vacuum spinning (Kerr) black holes. In particular, we find that the loss of a Carter-like constant leads to geodesic non-integrability and the onset of chaos. This manifests through the formation of resonant islands and chaotic layers around transient orbital resonances in phase space--features that are otherwise completely absent in integrable Kerr geodesics. Resonant islands, which are extended, non-zero volume regions in phase space, encapsulate periodic orbit points. Non-integrability dictates that all geodesics inside the resonant island share the periodicity of the resonance. Thus, the lifespan of resonances around non-Kerr objects can be significantly enhanced beyond the predicted lifetime of Kerr resonances. Consequently, these effects can leave distinct imprints on gravitational-wave signals, with significant implications for gravitational-wave modeling and parameter inference of astrophysical extreme-mass-ratio inspirals.