Accretion Disk Perturbations and Their Effects on Kerr Black Hole Superradiance and Gravitational Atom Evolution

Abstract

Kerr black hole (BH) superradiance can form gravitational atoms and produce characteristic gravitational-wave signals, providing a probe of ultralight bosons and dark matter. In realistic systems, accretion-disk gravity can shift energy levels and mix states, modifying the effective superradiant growth. We model the disk as a weak external perturbation via a multipole expansion and derive an effective three-level Hamiltonian for the $n=2$ subspace $\{\ket{211},\ket{210},\ket{21-1}\}$ in the weak-coupling regime. The leading disk effect is the quadrupolar ($\ell_d=2$) tidal field, whose symmetries fix the selection rules: axisymmetry gives only diagonal shifts, equatorial nonaxisymmetry activates $Δm=\pm2$ mixing ($\ket{211}\leftrightarrow\ket{21-1}$), and breaking equatorial reflection opens $Δm=\pm1$ couplings involving $\ket{210}$. As illustrations, a transient equatorial $m=2$ spiral wave drives the resulting two-level system and can suppress or quench superradiance by populating a decaying mode, while a quasi-static warp produces full three-level mixing and can generate narrow ``growth gaps'' near accidental near-degeneracies, with the same static reshuffling also allowing enhancement when weight shifts toward the growing mode. These findings demonstrate that accretion disk perturbations are a crucial environmental factor in determining the dynamics of BH superradiance and the evolution of boson clouds, thereby providing a more reliable theoretical basis for assessing the detectability of ultralight bosons in realistic astrophysical settings.

Figures (8)

• Figure 1: Partial energy levels of the gravitational atom. The states $\ket{211}$, $\ket{210}$, and $\ket{21-1}$ are shown together with the nearby $\ket{200}$ level. Green solid lines denote growing states, red dashed lines denote decaying states. $\Delta E$ is the energy difference between $\ket{210}$ and $\ket{200}$, while $\epsilon_h$ denotes the splitting between adjacent magnetic quantum numbers $m$ within the $\ket{21m}$ subspace.
• Figure 2: Schematic configuration of the BH, the boson cloud, and a geometrically thin accretion disk. The cloud is characterized by a typical radius $r_c$, while the disk extends from the inner radius $r_{\rm in}$ to the outer radius $r_{\rm out}$. The hierarchy $r_{\rm in}<r_c<r_{\rm out}$ illustrates that the cloud typically resides well inside the bulk of the disk. For example, for $\tilde{a}=0.9$ and $\alpha=0.1$, one has $r_{\rm in}=r_{\rm isco}\approx 2.3M$, $r_c\simeq 400M$, and a representative outer radius $r_{\rm out}\sim 10^5M$.
• Figure 3: Surface-density profile of an $m=2$ spiral density wave with a Gaussian envelope. The $m=2$ mode produces two spiral arms, while the Gaussian envelope localizes the perturbation radially. The pattern corotates and the wave packet drifts outward.
• Figure 4: Scan in the $(\alpha,\Sigma_0)$ plane for an equatorial $m=2$ spiral perturbation with a Gaussian envelope. (a) Steady-state effective growth rate $\Gamma_{\mathrm{eff}}(\infty)$; (b) growth-factor ratio $\chi\equiv \Gamma_{\mathrm{eff}}(\infty)/\Gamma_{\mathrm{eff}}(0)$, where $\chi\simeq1$ indicates negligible mixing, $0<\chi<1$ suppression, and $\chi<0$ quenching of superradiance. Crosses mark $(\alpha,\Sigma_0)=(0.032,6\times10^{-13})$ (red), $(0.038,4\times10^{-13})$ (blue), and $(0.044,3\times10^{-13})$ (green) (see Fig. \ref{['evolution']}).
• Figure 5: Time evolution of the two-level probabilities for several representative $(\alpha,\Sigma_0)$ choices (crosses in Fig. \ref{['Gamma']}). Solid and dashed curves show $|c_g(t)|^2$ and $|c_d^{(2)}(t)|^2$, respectively.