Trails of clouds in binary black holes

TL;DR

Ultralight bosons around rotating black holes form gravitational atoms whose clouds interact with a binary companion. The authors develop a worldline EFT framework to model generic binaries on eccentric and inclined orbits, capturing resonant and non-resonant cloud–orbit transitions beyond balance laws. They demonstrate floating and sinking phenomena, fixed points in eccentricity and obliquity, and a rich set of misalignment dynamics, extending previous results to comparable-mass binaries. The work yields concrete predictions for in-band and off-band signatures that could be probed by LISA, Cosmic Explorer, and the Einstein Telescope, enabling new tests of ultralight particles.

Abstract

Superradiant instabilities of rotating black holes can give rise to long-lived bosonic clouds, offering natural laboratories to probe ultralight particles across a wide range of parameter space. The presence of a companion can dramatically impact both the cloud's evolution and the binary's orbital dynamics, generating a trail of feedback effects that require detailed modelling. Using a worldline effective field theory approach, we develop a systematic framework for binaries on generic (eccentric and inclined) orbits, capturing both resonant and non-resonant transitions without relying solely on balance laws. We demonstrate the existence of ``co-rotating'' floating orbits that can deplete the cloud prior to entering the detector's band, triggering eccentricity growth towards a sequence of fixed points. Likewise, we show that ``counter-rotating'' orbits can also deplete the cloud, driving (unbounded) growth of eccentricity. Furthermore, we uncover novel features tied to orbital inclination. Depending on the mass ratio, equatorial orbits can become unstable, and fixed points may arise not only for aligned or anti-aligned configurations but, strikingly, also at intermediate inclinations. We derive flow equations governing spin-orbit misalignment and eccentricity and identify distinctive signatures that can reveal the presence of boson clouds in the binary's history, as well as key features of possible in-band transitions. These results refine and extend earlier work, yielding a more faithful description of the imprints of ultralight particles in gravitational-wave signals from binary black holes, signatures that are within reach of future detectors such as LISA, Cosmic Explorer, and the Einstein~Telescope.

Figures (20)

• Figure 1: Euler-angle rotation $\frak{R}(\bm{n},\bm{\hat{L}})$ from the reference frame, defined by the fixed axis $\bm{n}$, to the (non-inertial) orbital frame, whose $z$-axis is aligned with the orbital angular momentum $\bm{L}$, while the $y$-axis points in the direction orthogonal to the periapsis, with the orbital elements $\mathbb{E}$ indicated.
• Figure 2: Evolution of the orbital frequency [ left] and eccentricity [ right] through the $(k=-1)$$\ket{544} \to \ket{533}$ transition, with $(q,\alpha)_\mathrm{sat}=(1,0.3)$, $(N_\mathrm{c}/M^2)_\mathrm{sat}=0.13$, $e(\Omega^\mathrm{sat}_{1,-1})=0.05$, and in the weak-decay regime. Numerical solution (solid black), floating period and eccentricity growth [via \ref{['eq:flt_time']},\ref{['eq:v_flt']}] (red, dot-dashed), and standard radiation-reaction (RR) vacuum evolution (cyan, dotted).
• Figure 3: Time evolution of the orbital frequency [ left] and frequency evolution of the eccentricity [ right] through the $\ket{322} \to \ket{311}$ transition, with $(q,\alpha)_\mathrm{sat}=(0.1,0.22)$, $(N_\mathrm{c}/M^2)_\mathrm{sat}=0.33$, and $e(\Omega^\mathrm{sat}_{g,k})=0.1$. Numerical solution (solid black) and radiation-reaction vacuum evolution (cyan, dotted) are shown in both cases for the co-rotating case ($g=1,k=-1)$. We also illustrate the growth of eccentricity for the $k=-2$ counter-rotating overtone (blue).
• Figure 4: Time evolution of the orbital frequency [ left] and eccentricity [ right] during a sinking (early Bohr) transition $\ket{211} \to \ket{54-4}$ on the counter-rotating orbit for $(g=-5,k=-1,0,1)$ overtones (dashed purple, solid red, and dot-dashed green), with $(q,\alpha)_\mathrm{sat}=(0.05,0.1)$, $(N_\mathrm{c}/M^2)_\mathrm{sat}=0.33$, $e(\Omega^\mathrm{sat}_{g,k})=0.05$, respectively. We also plot the averaged $k=-1$ resonance (black) and the corresponding vacuum (dotted, cyan) evolution.
• Figure 5: Eccentricity/obliquity $\{e,\beta\}$ flow for degenerate overtones $\Omega^{(ab)}_{(g,k|d)} = (m/d) \, \Omega^{(ab)}_0$ in the limit $(b_{d}/w_d)^{(ab)} \gg 1$ [via \ref{['eq:e_i_emri']}] for $l=2$ and: $(m,d)=(2,2)$ [upper left], $(2,3)$ [upper right], $(1,2)$ [lower left] and $(1,3)$ [lower right]. Fixed points are indicated by red dots, and (non-equatorial) separatrices by purple and green dashed curves. The strongest (eccentric) overtone on the left diagrams corresponds to the main one, at $k=0$, while on the right the strongest one is at $k=-1$. The lower panel exhibits the non-equatorial fixed points at $(e,\beta_\mathrm{cr}) = (0, \pi/3)$ [ left; cf. \ref{['eq:beta_circ_emri']}] and $(0.48, \pi/4)$ [ right].