Ring formation from black hole superradiance through repeated particle production on bound orbits

TL;DR

BH superradiance converts rotational energy into a growing ultralight-boson cloud with a hydrogenlike spectrum. Introducing a second axion coupled by $V_{ m int}=\frac{1}{2}\lambda\phi\chi^2$ reveals a novel mechanism where resonant production of a heavier field $\chi$ occurs primarily at the cloud’s toroidal peak, and bound $\chi$ particles on quasi-circular orbits undergo repeated resonance crossings, leading to staircase amplification and the formation of a stable ring embedded in the cloud. The ring-saturation dynamics show that the final ring mass is set by the hierarchy $\mu_\phi\ll\mu_\chi$, giving $\frac{M_\chi}{M_{\rm cloud}}\approx 0.6\left(\frac{\mu_\phi}{\mu_\chi}\right)^2$, and that backreaction halts growth. The work highlights a bound-state production channel with potential observational consequences in gravitational waves and binary dynamics, distinct from scenarios with quartic couplings which instead produce an escaping flux.

Abstract

Ultralight bosonic fields around a rotating black hole can extract energy and angular momentum through the superradiant instability and form a dense cloud. We investigate the scenario involving two scalar fields, $φ$ and $χ$, with a coupling term $\frac{1}{2}λφχ^2$, which is motivated by the multiple-axion framework. The ultralight scalar $φ$ forms a cloud that efficiently produces $χ$ particles nonperturbatively via parametric resonance, with a large mass hierarchy, $μ_χ\gg μ_φ$. Rather than escaping the system as investigated by previous studies, these $χ$ particles remain bound, orbiting the black hole. Moreover, the particle production occurs primarily at the peak of the cloud's profile, allowing $χ$ particles in quasicircular orbits to pass repeatedly through resonant regions, leading to cumulative amplification. This selective process naturally forms a dense ring of $χ$ particles, with a mass ratio to the cloud fixed by $(μ_φ/μ_χ)^2$. Our findings reveal a novel mechanism for generating bound-state particles via parametric resonance, which also impacts the evolution of the cloud. This process can be probed through its imprint on binary dynamics and its gravitational-wave signatures.

Figures (5)

• Figure 1: The radial motion of a particle in a numerically solved, stable quasicircular orbit, plotted over one orbital period with benchmark parameters ($\mu_\chi/\mu_\phi=100$, $\lambda\Phi_0/\mu_\chi^2=1$, and $\alpha=0.15$). Initial conditions are $r_i=2r_0$, $\theta_i=\pi/2$, and $\varphi_i=\pi+0.1$, modeling the particle initially produced near the resonance region with angular velocity $\dot{\varphi}_i\simeq 1.24\alpha$ and initial momenta $k\simeq 0.2\mu_\chi$. The radial coordinate $r$ oscillates with only small deviations around the initial radius $2r_0$, confirming the stability of the orbit.
• Figure 2: The illustration of the cumulative amplification mechanism for a trapped $\chi$ particle. This figure schematically shows the "staircase" growth of a trapped mode's occupation number $\abs{n_{{k}_{\star}}}$ as it moves along its quasicircular orbit. The smooth, horizontal segment (black line) represents the particle's motion between resonant regions, where its amplitude remains constant. The sharp vertical jumps in the blue line represent brief, intense amplification of the occupation number $n_{{k}_{\star}}$ as the particle crosses resonance planes (the gray planes), occurring each time $\phi$ oscillates to its minimum (the phase of $\phi$ is shown by the red line).
• Figure 3: A schematic of the final cloud-ring configuration. The larger, transparent orange structure shows the toroidal density profile of the primary superradiant cloud of the $\phi$ axion (in the (211) state). Embedded within this cloud, near its density peak at $r=2r_0=2r_g/\alpha^2$ is the stable secondary ring composed of $\chi$ particles (blue solid ring). This illustrates the saturated final state of the system, where the primary cloud coexists with the particle ring it has created.
• Figure 4: Particle trajectory in the equatorial plane for the quartic interaction scenario. The parameters used for this simulation are $\mu_\chi/\mu_\phi=100$, $g^2\Phi_0^2/\mu_\chi^2=10^4$, and $\alpha=0.1$.
• Figure 5: Evolution of particle velocity over time for the quartic interaction scenario. The dimensionless time is defined by $z=\Omega t$ (see Appendix \ref{['app:stability']}). The parameters are the same as above.