Binary-Perturbed Gravitational Atoms

• Binary-perturbed gravitational atoms are black-hole boson cloud systems disrupted by a binary companion's tidal forces, enabling resonant transitions, ionization, and accretion.
• The system is modeled using hydrogenic bound states around a Kerr black hole, where the companion induces time-dependent perturbations that impact cloud evolution and gravitational-wave signatures.
• Ionization, dynamical friction, and resonant backreaction provide measurable effects on binary inspirals, offering a unique probe into ultralight bosons and black-hole spin extraction.

Binary-perturbed gravitational atoms are black-hole–boson-cloud systems in which a compact binary companion tidally drives the cloud away from its isolated superradiant evolution. In the nonrelativistic regime, the cloud is described by hydrogenic bound states around a Kerr black hole, but the companion’s time-dependent potential introduces three qualitatively distinct channels: resonant bound-to-bound transitions, bound-to-continuum ionization, and—when the companion is itself a black hole—accretion of cloud material onto the secondary. These processes backreact on the orbit and on the cloud occupation numbers, producing characteristic modifications of the emitted gravitational waves and, in some regimes, a direct gravitational-wave signal from the cloud transition itself (Baumann et al., 2021, Kyriazis et al., 23 Mar 2025, Tomaselli, 2024).

1. Gravitational-atom framework

A gravitational atom consists of a Kerr black hole of mass $M$ and dimensionless spin $\tilde a$, surrounded by a bosonic cloud generated by superradiance. For a boson of mass $\mu$, the relevant coupling is the gravitational fine-structure constant

$\alpha \equiv \mu M \ll 1,$

and superradiance requires

$$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$

In this regime, the scalar field can be written in terms of a slowly varying envelope $\psi$ obeying

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$

so the cloud admits hydrogenic bound states $\ket{n\ell m}$ with wavefunctions

$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$

and energies

$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$

up to higher-order corrections in $\tilde a$0 and spin-dependent splittings (Kyriazis et al., 23 Mar 2025, Tomaselli, 2024).

The cloud’s characteristic size is the gravitational analog of a Bohr radius,

$\tilde a$1

and the cloud mass can grow to $\tilde a$2 in the fastest mode $\tilde a$3 (Tomaselli et al., 2023). Superradiant growth of an initial $\tilde a$4 cloud occurs on a timescale

$\tilde a$5

and clouds with $\tilde a$6–$\tilde a$7 can form around astrophysical black holes for $\tilde a$8–$\tilde a$9 (Baumann et al., 2021). The Kerr spin saturates at the critical value

$\mu$0

which fixes the endpoint of superradiant extraction in the hydrogenic regime (Kyriazis et al., 23 Mar 2025).

2. Binary perturbation and tidal couplings

When the black hole–cloud system belongs to a binary, a companion of mass $\mu$1 on an orbit with separation $\mu$2, phase $\mu$3, and angular frequency $\mu$4 generates a time-dependent tidal perturbation. In a Newtonian multipole expansion, the companion potential can be written as

$\mu$5

which enters the Schrödinger Hamiltonian as $\mu$6 (Tomaselli, 2024). For $\mu$7, the leading piece is quadrupolar,

$\mu$8

and this leading-order perturbation mixes levels satisfying $\mu$9 and $\alpha \equiv \mu M \ll 1,$0 (Kyriazis et al., 23 Mar 2025).

Matrix elements between cloud states are overlap integrals of radial hydrogenic functions and spherical harmonics. In the general expansion,

$\alpha \equiv \mu M \ll 1,$1

where the Fourier coefficients $\alpha \equiv \mu M \ll 1,$2 depend slowly on $\alpha \equiv \mu M \ll 1,$3 and encode the strength of the $\alpha \equiv \mu M \ll 1,$4 harmonic (Tomaselli, 2024). The angular structure imposes the selection rules

$\alpha \equiv \mu M \ll 1,$5

while, in the quadrupolar approximation, the off-diagonal two-state coupling takes the dressed form

$\alpha \equiv \mu M \ll 1,$6

(Kyriazis et al., 23 Mar 2025).

This perturbative formulation provides a single framework for both discrete resonances and ionization. Bound-state mixing follows from matrix elements between discrete levels; ionization follows from the same tidal operator acting between a bound state and continuum states $\alpha \equiv \mu M \ll 1,$7 (Baumann et al., 2021, Tomaselli, 2024).

3. Resonant bound-state transitions

A resonant transition occurs when the orbital driving matches a cloud level splitting. For two bound states $\alpha \equiv \mu M \ll 1,$8 and $\alpha \equiv \mu M \ll 1,$9, the resonance condition is

$$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$0

or, in the notation used for generic harmonics,

$$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$1

with $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$2 (Baumann et al., 2021, Tomaselli et al., 2024). In the two-level approximation, one passes to a dressed frame and obtains a Landau–Zener Hamiltonian. For quasi-circular inspiral with $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$3, one form is

$$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$4

where $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$5 is the decay rate of the final state into the hole (Kyriazis et al., 23 Mar 2025). In the limit $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$6, the standard transition probability is

$$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$7

equivalently $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$8 in the notation of the resonant-history analysis (Kyriazis et al., 23 Mar 2025, Tomaselli et al., 2024).

The resonance radius has a characteristic scaling with $$\omega < m\,\Omega_H, \qquad \Omega_H \equiv \frac{\tilde a}{2r_+}, \qquad r_+=M+\sqrt{M^2-a^2}.$$9,

$\psi$0

with

$\psi$1

depending on whether the transition changes $\psi$2, $\psi$3, or $\psi$4 (Tomaselli et al., 2024). This hierarchy organizes the chronological sequence of resonances encountered during inspiral.

Generic eccentricity and inclination qualitatively modify whether a resonance operates. The resonant-history analysis introduces a backreaction parameter

$\psi$5

and shows that a floating resonance starts only if

$\psi$6

For eccentricity $\psi$7, the effective chirp rate is modified by

$\psi$8

and the critical eccentricity satisfies $\psi$9; for inclination $$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$0, the coupling carries a Wigner-$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$1 factor $$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$2, which can suppress the resonance below threshold (Tomaselli et al., 2024). If the upper state has a nonzero horizon-decay rate $$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$3, the resonance can also break once

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$4

These results establish that binary-perturbed gravitational atoms are not described by a universal single-resonance picture. Whether a resonance is adiabatic, floating, broken by decay, or never activated depends on the sequence of crossings, the orbital geometry, and the cloud backreaction (Tomaselli et al., 2024).

4. Ionization, dynamical friction, and accretion

Ionization is the bound-to-continuum channel in which the companion supplies enough orbital energy to unbind cloud quanta. In harmonic decomposition, the resonance condition for a continuum mode of wavenumber $$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$5 is

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$6

Integrating out continuum states in the Markov approximation yields the bound-state depletion law

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$7

and the ionized power

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$8

(Baumann et al., 2021). The same process can be derived from first-order perturbation theory and Fermi’s Golden Rule in the companion frame (Tomaselli, 2024).

For typical parameters,

$$i\partial_t\psi=\left[-\frac{\nabla^2}{2\mu}-\frac{\alpha}{r}\right]\psi,$$9

the orbital power lost to ionization satisfies

$\ket{n\ell m}$0

at separations $\ket{n\ell m}$1, so ionization can dominate over gravitational-wave radiation (Baumann et al., 2021). The ionization spectrum exhibits sharp jumps when $\ket{n\ell m}$2, producing threshold features or “kinks” that encode the cloud energy spectrum. In gravitational-wave frequency, the thresholds occur at

$\ket{n\ell m}$3

or equivalently

$\ket{n\ell m}$4

depending on the notation adopted (Tomaselli et al., 2023, Tomaselli, 2024).

A central result of the later literature is that ionization is equivalent to dynamical friction. The classical collisionless expression,

$\ket{n\ell m}$5

matches the ionization power up to $\ket{n\ell m}$6 factors, so the cloud-induced backreaction on the orbit should be identified as dynamical friction (Tomaselli et al., 2023). In the small-$\ket{n\ell m}$7 limit,

$\ket{n\ell m}$8

which explains why the effect is strongest when the companion traverses the cloud (Tomaselli et al., 2023). On open orbits, the additional energy dissipation can increase the dynamical-capture cross section by more than an order of magnitude, with typical $\ket{n\ell m}$9–$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$0 in the regime where cloud losses dominate (Tomaselli et al., 2023).

If the companion is a black hole, cloud matter can also be accreted. In the nonrelativistic “fuzzy” regime $$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$1 and $$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$2, the mass-growth rate is

$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$3

with horizon area $$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$4 and local cloud density $$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$5 (Baumann et al., 2021). An equivalent particle-language expression is

$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$6

and over the inspiral this accretion can increase the companion mass by $$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$7 or more (Baumann et al., 2021).

5. Backreaction on the binary and resonant histories

The orbit evolves under a combined energy budget rather than the vacuum quadrupole formula alone. In the Newtonian circular approximation,

$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$8

with

$$\psi_{n\ell m}(t,\mathbf r)=e^{-i(\omega_{n\ell m}-\mu)t}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$9

(Monica et al., 30 Mar 2025). Realistic waveform generation therefore requires integrating the coupled ODEs for the orbital separation and the cloud and black-hole masses, rather than adding a perturbative phase correction to an otherwise vacuum inspiral (Monica et al., 30 Mar 2025).

The different channels act on different timescales. Resonant transitions occur on

$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$0

while ionization turns on when

$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$1

and then depletes the cloud on

$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$2

which can be much shorter than the gravitational-wave chirp time

$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$3

(Baumann et al., 2021). This scale separation is one reason ionization often dominates the late cloud evolution once the companion enters the Bohr region.

Ionization and resonances affect orbital geometry differently. Ionization quickly circularizes the binary but barely affects the inclination angle: numerical solutions give $$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$4–$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$5 when the semimajor axis is comparable to the cloud size, whereas $$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$6–$$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$7 and the total inclination change is $$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$8 (Tomaselli et al., 2023). By contrast, resonant backreaction at generic eccentricity and inclination can produce dramatic changes in both $$E_{n\ell m}\simeq \mu\left(1-\frac{\alpha^2}{2n^2}\right),$$9 and $\tilde a$00, and the system’s long-term history bifurcates into two regimes (Tomaselli et al., 2024).

In the first regime, “cloud destruction,” the first hyperfine or fine resonance is adiabatic and the cloud is fully transferred into a decaying state, after which the binary continues essentially as a vacuum inspiral but with finite kicks in eccentricity and inclination. In the second regime, “cloud survival,” near-counter-rotating systems or binaries entering too late for early resonances to become adiabatic retain the cloud until the Bohr region, where continuous ionization with $\tilde a$01 produces a universal chirp together with nonadiabatic “sinking” jumps at each Bohr-resonant frequency. The associated phase jump satisfies

$\tilde a$02

which is stated to be well above the LISA detectability threshold $\tilde a$03 (Tomaselli et al., 2024).

6. Gravitational-wave phenomenology and detectability

The most immediate gravitational-wave imprint is dephasing of the binary inspiral. Because the orbit loses energy through ionization and, when applicable, through companion accretion, the gravitational-wave phase acquires an extra contribution

$\tilde a$04

which is dominated by ionization when $\tilde a$05 (Baumann et al., 2021). Each ionization kink changes the slope of $\tilde a$06 at the corresponding threshold frequency, yielding sharp spectroscopic features absent in vacuum templates (Baumann et al., 2021). In frequency-domain waveform models built from a reference IMR template, the environment-induced dephasing is

$\tilde a$07

and amplitude corrections are typically subdominant and often neglected (Monica et al., 30 Mar 2025).

A second, distinct signal is the gravitational radiation emitted directly by a resonantly driven cloud transition. In the two-level approximation, the time-dependent cloud density contains an interference term $\tilde a$08 that sources a mass quadrupole. The resulting strains at distance $\tilde a$09 are

$\tilde a$10

with

$\tilde a$11

(Kyriazis et al., 23 Mar 2025). In the stationary-phase approximation, the Fourier spectrum is peaked at

$\tilde a$12

with width $\tilde a$13 in the small-$\tilde a$14 regime, and these observables can be combined with inspiral information to infer $\tilde a$15, $\tilde a$16, the cloud mass fraction, and thus the boson mass $\tilde a$17 and black-hole spin $\tilde a$18 (Kyriazis et al., 23 Mar 2025).

Forecasts for detector sensitivity depend strongly on mass scale. For LISA-oriented systems, parameters

$\tilde a$19

yield ionization kinks and an accretion-induced phase drift that are potentially observable with $\tilde a$20, and their combined detection would allow $\tilde a$21 to be measured to percent-level accuracy (Baumann et al., 2021). For next-generation ground-based detectors, Fisher forecasts for the Einstein Telescope indicate that binaries with

$\tilde a$22

can accumulate $\tilde a$23, with mismatches $\tilde a$24 and relative uncertainties on $\tilde a$25 and $\tilde a$26 at the $\tilde a$27–$\tilde a$28 level in Fisher analysis (Monica et al., 30 Mar 2025).

At much higher frequencies, primordial-black-hole gravitational atoms shift the discussion to the MHz–GHz band. A benchmark binary with

$\tilde a$29

has resonant orbital frequency $\tilde a$30, within the ADMX Run-1 band $\tilde a$31, but its characteristic strain is only $\tilde a$32, compared to $\tilde a$33; reaching the latter would require $\tilde a$34 (Su et al., 1 Apr 2026). That analysis concludes that binary-driven high-frequency transients are generically too weak and too rare at astrophysical distances for current experiments, although isolated annihilation signals may be more promising targets (Su et al., 1 Apr 2026).

A recurrent misconception is that binary perturbations are exhausted by Landau–Zener bound-state resonances. The literature instead supports a broader picture: ionization can dominate the energy budget, accretion can significantly change the companion mass, and the cloud state entering an observational band depends on the full resonant history, including orbital geometry and nonlinear backreaction (Baumann et al., 2021, Tomaselli et al., 2024). Under that more complete description, binary-perturbed gravitational atoms function as spectroscopic systems whose orbital and direct gravitational-wave signatures jointly probe ultralight bosons, black-hole spin extraction, and environmental modifications of compact-binary inspiral.