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

TL;DR

This work addresses how realistic astrophysical environments influence extreme-mass-ratio inspirals by constructing fully relativistic rotating black holes embedded in Hernquist-type halos and studying geodesic motion on these backgrounds. It shows that environmental backreaction can break the Carter symmetry, producing non-integrable geodesics with resonant islands and thin chaotic layers that can imprint on gravitational-wave signals. The width of the dominant 2/3 resonance grows with both the black hole spin and halo compactness, implying stronger chaos and potentially large GW dephasing or glitches in EMRIs. These findings highlight the necessity of environment-aware, fully relativistic EMRI modeling for accurate GW data analysis and motivate extending to radiation reaction and perturbative treatments for non-vacuum spacetimes in the LISA era.

Abstract

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.

Figures (3)

• Figure 1: Left: Equatorial Poincaré map of generic geodesics with fixed $E/\mu=0.9$ and $L_z/\mu=3M_{\rm BH}$ and varying initial position $r(0)/r_h$, where $r=r_h$ is the radius of the event horizon in quasi-isotropic coordinates. The rotating BH embedded in the Hernquist-type environment, where the geodesics are evolved, has spin $J/M_{\textrm{BH}}^2\simeq0.996$, halo compactness $M_\textrm{halo}/a_0\simeq10^{-1}$, and halo mass $M_\textrm{halo}\simeq 10 M_\textrm{BH}$. The black curves formed around the central fixed point of the map are KAM curves that arise from orbits with irrational ratios $\omega_r/\omega_\theta$. On the other hand, the blue and red curves designate resonant islands, with rational frequency ratios $\omega_r/\omega_\theta=2/5, \,2/3$, respectively. These islands encapsulate their corresponding stable periodic points. In turn, the green formation that appears in the map is a chaotic zone, associated with the thin chaotic layer that encircle the resonant islands with rational ratio $\omega_r/\omega_\theta=1/2$ (not shown here). Right: Same as left, zoomed-in region at the strong-field regime. The red and blue islands depicted surround the stable periodic points of the resonance $\omega_r/\omega_\theta=2/3$ and $2/5$, respectively, while the green points form a chaotic layer around the $\omega_r/\omega_\theta=1/2$ resonant islands (not shown here).
• Figure 2: Left: Rotation curve of geodesics with fixed $E/\mu=0.9$, $L_z/\mu=3M_{\textrm{BH}}$ and varying initial position $r(0)/r_h$, resulting from an equatorial surface of section of successive geodesics. The rotating BH embedded in the Hernquist-type environment, where the geodesics are evolved, has spin $J/M_{\textrm{BH}}^2\simeq0.996$, halo compactness $M_\textrm{halo}/a_0\simeq10^{-1}$, and halo mass $M_\textrm{halo}\simeq 10 M_\textrm{BH}$. The red box corresponds to the zoomed-in region of a resonant island, where a plateau is formed at $\nu_\vartheta=2/3$. Right: Same as left configuration in the phase space vicinity of the $2/5$-resonant island. A plateau is formed exactly at $\nu_\vartheta=2/5$.
• Figure 3: Left: Width of the $2/3$-resonant islands formed in the phase space of BHs in matter halos with varying spin parameter $J/M_{\textrm{BH}}^2$ and fixed compactness $M_\textrm{halo}/a_0\simeq10^{-1}$, with $M_\textrm{halo}\simeq 10 M_\textrm{BH}$. The island widths result from geodesic evolutions around the aforementioned configurations, with $E/\mu=0.9$ and $L_z/\mu=3M_{\textrm{BH}}$. The island width has been normalized with respect to the event horizon radius $r=r_h$ of each configuration in quasi-isotropic coordinates. Right: Same as left, but with fixed BH spin $J/M_{\textrm{BH}}^2\simeq 0.8$ and varying compactness $M_\textrm{halo}/a_0$, with $M_\textrm{halo}\simeq 10 M_\textrm{BH}$. For both panels, the red dots correspond to the nine different BH configurations analyzed, while the dashed black lines correspond to crude interpolations of these values.