Self-Gravity in Superradiance Clouds: Implications for Binary Dynamics and Observational Prospects

Abstract

Spinning black holes could produce ultralight particles via the superradiance instability. These particles form a dense cloud around the host black hole, introducing new opportunities for the detection of ultralight new physics. When the black hole is part of a binary system, the binary can trigger transitions among different states of the cloud configuration. Such transitions backreact on the orbital dynamics, modifying the frequency evolution of the emitted gravitational waves. Based on this observation, black hole binaries were proposed as a way to test the existence of ultralight particles. We investigate the effects of the self-gravity of the cloud on the orbital evolution and on the gravitational wave emission. We find that cloud self-gravity could lead to a density-dependent modification of the energy levels of ultralight particles and that it could alter the order of hyperfine energy levels. The crossing of hyperfine levels prevents binaries from triggering resonant hyperfine transitions and allows them to approach radii that could trigger resonant transitions of fine levels. We study the implications of these findings, especially in the context of future space-borne gravitational wave observatory, the Laser Interferometer Space Antenna (LISA). For quasi-circular, prograde and equatorial orbits, we find that LISA could probe ultralight particles in the mass range $10^{-15}\,{\rm eV} \, - \, 10^{-13}\, {\rm eV}$ through gravitational wave observations.

Figures (16)

• Figure 1: Parameter space showing current constraints and the region in which ultralight particles can be probed with LISA for a total observational time span $T_{\rm obs} = 4\ {\rm yr}$. Shaded area indicates regions where LISA is sensitive to ultralight particles via observations of GWs emitted at the hyperfine resonance $|322\rangle \leftrightarrow |320\rangle$ (blue) and at the fine resonance $|322\rangle \leftrightarrow |31-1\rangle$ (purple). These regions are based on the computation of fitting factor, which will be discussed in Sec. \ref{['sec:observational_target']}. The region shaded in lighter blue is where our approximation of neglecting off-diagonal matrix element of the self-gravity breaks down (see Sec. \ref{['sec:off_diagonal']}). The fine resonance can be reached due to the level crossing induced by the self-gravity of the cloud. The black contours show the horizon distance at which LISA can observe emitted GWs with ${\rm SNR}=5$, while the red contours shows the mass of the spinning black hole $M_1$. We consider only quasi-circular, prograde, equatorial orbits and assume the mass of secondary object $q = M_2/M_1 =0.05$ for the hyperfine transition and $q =0.02$ for the fine transition. Constraints from black hole spin-down are overlaid as vertical gray bands Arvanitaki:2014wvaHoof:2024qukAswathi:2025nxaCaputo:2025oap.
• Figure 2: Cloud mass fraction $q_c = M_c/M_1$ for $t_{\rm sys} =100\,{\rm Myr}$ and initial BH spin $a_*^i=0.9$. The right panel shows the cloud mass fraction $q_c$ for $|211\rangle$ and $|322\rangle$ states. We only show the region with $q_c > 10^{-5}$. The upper boundary of the contours is due to the annihilation of the cloud into gravitational waves, while the lower boundary arises because the age of the system $t_{\rm sys}$ is too short for the superradiance instability to develop. The star corresponds to a benchmark point for which the cloud evolution is studied as a function of time in Figure \ref{['fig:qc_max_100Myr_2']}. The left panel shows the cloud mass fraction of $|322\rangle$ at $t_{\rm sys} = 100\,$Myr as a function of $\alpha_i$ for $\mu = 10^{-15}\,\textrm{--} \, 10^{-12}\ {\rm eV}$. The red dashed line shows the maximum achievable cloud mass without the annihilation of bosons into gravitational waves. The black line shows the behavior of $q_c^{\rm max}\propto \alpha^2$ for small fine structure constants. See Appendix \ref{['app:cloud_mass']} for details on the cloud mass computation.
• Figure 3: (Top) a schematic picture describing the sequence of important orbital resonances with an initial $|322\rangle$ cloud. The orange band around $|322\rangle\leftrightarrow |300\rangle$ denotes the radii around which the mixing with $|300\rangle$ significantly backreacts to the orbit and a three-level description is necessary. This will be discussed in Section \ref{['sec:mixing']}. (Bottom) the orbital dynamics at the resonance $|322\rangle \leftrightarrow |320\rangle$ without self-gravity corrections. Compared to the evolution without the cloud (dashed line), the binary hardens much more slowly. The green line denotes the evolution obtained in the two-level approximation, while the dark blue line is obtained in a three-level system, including $|300\rangle$. As this hyperfine transition is adiabatic, the orbit almost completely converts $|322\rangle \to |320\rangle$ and $|320\rangle$ decays subsequently. By the time the orbit reaches to orbital separations that can trigger fine transitions such as the $|322\rangle\leftrightarrow|31-1\rangle$ resonance, the entire cloud has disappeared; this conclusion will be altered when the self-gravity correction is included. Here $\Omega_0$ is the resonance frequency computed with the parameters at the beginning of the evolution and $\bar{t} = [\Omega(d\Omega/dt)_{\rm GW}^{-1}]|_{\Omega_0}$ is the typical evolution time scale for the orbit due to the GW emission.
• Figure 4: The energy level difference between the $|322\rangle$ state and a few other states. The level crossing occurs for the $|322\rangle \leftrightarrow |320\rangle$ transition for $\mu = 10^{-13}\ {\rm eV}$ around $\alpha \sim 0.31$. Fine transition levels are affected at most at 30%. We choose the value of $q_c$ with an initial spin parameter $a_* = 0.9$ and $t_{\rm age} =100\,{\rm Myr}$, and use the non-relativistic spectrum for this result. Dashed lines show the level spacing without the self-gravity corrections.
• Figure 5: (Left) the orbital dynamics at the hyperfine $|322\rangle \leftrightarrow |320\rangle$ resonance. All parameters are chosen the same as in Figure \ref{['fig:floating']}. The self-gravity correction is included. As the effective level splitting between these two states changes its sign, a prograde orbit can no longer trigger the resonant transition with them. The cloud still depletes in the 3-level analysis, which is due to the large decay width of the non-superradiance $|300\rangle$ state. The above result suggests that the orbit can reach closer to the black hole, and trigger the resonant transition of the fine levels $|322\rangle \leftrightarrow |31-1\rangle$. (Right) the orbital dynamics around the fine transition $|322\rangle \leftrightarrow |31-1\rangle$. The difference between two-level and three-level approximation is noticeable. The mixing with $|300\rangle$ makes the orbit harden at a much slower rate well before the binary enters the resonance band of the $|322\rangle \leftrightarrow |31-1\rangle$ transition. If the cloud somehow survives by the time it enters the resonance band of the fine transition, there could be another period of evolution in which the binary exhibits a floating behavior as can be seen in the inset plot. For this result, we choose a smaller $q =0.03$. Here $\Omega_0$ is the resonance frequency of each level, computed at the beginning of the numerical evolution with the self-gravity corrections.