Gravitational Atoms and Black Hole Binaries

Giovanni Maria Tomaselli

REVIEW 2 major objections 7 minor 14 references

Gravitational Atoms and Black Hole Binaries

T0 review · 2 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The thesis claims that boson clouds formed by black hole superradiance alter binary inspirals so strongly that ionization, not gravitational radiation, can set the merger timescale, and that orbital resonances imprint preferred…

desk verdict A careful, honest PhD thesis that compiles the author's own published results; the physics is solid, but the quantitative claims rest on free-parameter clouds and the document adds no new result beyond those papers. read the letter →

arxiv 2412.12526 v1 pith:TR3TXRAU submitted 2024-12-17 gr-qc hep-phhep-th

Giovanni Maria Tomaselli This is my paper · ORCID

classification gr-qchep-phhep-th

keywords blackholesuperradiancebosoncloudgravitationalatomultralightbosonswavesbinaryinspiralionizationorbitalresonances

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis argues that an ultralight-boson cloud formed by black hole superradiance, called a gravitational atom, leaves measurable marks on the gravitational waves of a binary whose smaller black hole orbits the cloud. The central new effect is ionization: the companion's gravitational field unbinds bosons out of the cloud, and the energy taken from the orbit can exceed the power radiated in gravitational waves across much of the inspiral. The companion also drives resonant bound-to-bound transitions whose backreaction briefly stalls the inspiral and pushes the binary's eccentricity and inclination toward preferred fixed points. If the predictions hold, current and future detectors such as LISA, LIGO, Einstein Telescope, DECIGO, and TianQin can use binary waveforms to discover or constrain ultralight particles.

What carries the argument

The machinery is the gravitational-atom description: an ultralight scalar field around a Kerr black hole obeys a Schrödinger equation whose bound states are hydrogenic orbitals labeled by principal, angular, and azimuthal quantum numbers, with gravitational fine-structure constant $\alpha=\mu M$. The binary companion enters as a multipole-expanded Newtonian potential whose matrix elements between bound and continuum states obey angular selection rules and are expanded in Fourier components oscillating at orbital overtones. Ionization is computed with Fermi's Golden Rule, supplemented by a chirp-aware derivation that integrates out the continuum, to obtain the ionization rate, power $P_{\rm ion}$, and torque $\tau_{\rm ion}$. Resonances are treated as Landau-Zener transitions with adiabaticity parameter $Z=\eta^2/(|g|\gamma)$, and the backreaction on the orbit is encoded in a parameter $B$ that determines whether a resonance floats or sinks the inspiral.

What would settle it

Observe a massive black hole binary with LISA over a long inspiral and find that its frequency evolution matches the vacuum general-relativistic chirp within measurement error, for a system whose inferred parameters ($M\approx 10^4\,M_\odot$, spin near the superradiance threshold, $\alpha\approx 0.2$, companion mass ratio $q\approx 10^{-3}$) would give $P_{\rm ion}>P_{\rm gw}$ for a $|211\rangle$ cloud with $M_c/M\approx 0.01$; the absence of the predicted accelerated merger and of the ionization threshold discontinuities in the chirp would falsify the central claim for that boson mass.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dynamical interaction between a black hole binary and a superradiant boson cloud is not a small perturbation but often the dominant driver of the inspiral. Ionization, the analogue of the photoelectric effect in which the binary's time-varying gravitational potential kicks bound bosons into unbound states, removes energy from the orbit at a rate that can exceed the gravitational-wave luminosity for a wide range of separations, so the binary merges substantially sooner than in vacuum. In addition, the discrete hydrogen-like spectrum of the cloud makes resonant transitions sharp: Landau-Zener crossings occur at specific orbital frequencies, and their backreaction can create floating orbits where the inspiral is temporarily stalled while eccentricity and inclination are pushed toward fixed points. Together these effects create two kinds of signatures: direct ones in the waveform from ionization power and resonance dephasing, and indirect ones in the preferred residual eccentricities and inclinations that survive even if the cloud is destroyed before the observation.

Load-bearing premise

The quantitative predictions assume a pure, non-relativistic, self-interaction-free boson cloud whose mass $M_c$ is an independent parameter, set to the fiducial $M_c/M=0.01$; if scalar self-interactions trigger a bosenova that destroys the cloud before the inspiral, or if astrophysical clouds are far less massive, the large ionization power and resonance strengths that drive the claimed signatures will be reduced or absent.

Editorial extensions

If this is right

Ionization removes orbital energy faster than gravitational radiation over much of the inspiral, so binaries with clouds merge earlier than vacuum binaries with the same initial parameters.
Resonances act as filters on orbital elements, pushing eccentricity and inclination toward specific fixed points whose values are set by the cloud state and the resonance overtone number $g$.
The cloud enhances the dynamical-capture cross section by up to factors of tens to hundreds, raising the predicted merger rate for mass ratios around $q\sim 10^{-3}$ in dense environments.
Ionization depletes the cloud by an amount comparable to the companion's mass, so smaller clouds experience larger fractional mass loss during the inspiral.
The predicted signatures, ionization threshold steps, chirp dephasing, and fixed points in eccentricity and inclination, fall in the sensitivity bands of LISA, DECIGO, Einstein Telescope, and TianQin.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A key implicit consequence is that measuring a vacuum-like waveform in a candidate LISA event would not rule out ultralight bosons; it would only push the cloud mass below the value needed to dominate the inspiral, so searches should scan over $M_c/M$.
Because the ionization rate depends on the quantum numbers of the populated superradiant state, waveform fits could in principle identify which state the cloud occupied, effectively measuring the boson mass and the black hole's initial spin.
The framework naturally extends to distinguishing boson-cloud environments from other environmental effects such as dark-matter spikes or accretion disks, since ionization produces characteristic threshold steps in the chirp that other mechanisms do not.
A testable extension is to compute full waveform templates including the transient oscillations near ionization thresholds; their timescale $\gamma^{-1/2}$ would give a direct measurement of the effective chirp rate induced by the cloud.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read on Tomaselli's thesis. The one-line take: this is a well-organized PhD thesis that presents the author's own published work on gravitational atoms in black hole binaries, and it does that job well. The underlying physics—ionization, accretion, dynamical capture, and resonances—is internally consistent and the derivations are careful. If you want a single place to learn the subject, this is better than the five papers it is based on, because the background and notation are set up properly.

What is actually new: nothing beyond [1-5], and the thesis is upfront about that. Chapters 3-7 are a reordered compilation of the author's published results. That's not a flaw for a thesis, but it matters if someone cites this as the primary source.

The strengths: the treatment of ionization as dynamical friction (Sec 4.1) is conceptually illuminating; the scaling formulas (Sec 4.2) are practically useful; and the resonance backreaction analysis (Ch 5) is careful about where adiabaticity holds. The claim that ionization power can exceed GW luminosity during much of the inspiral is supported by explicit matrix-element calculations, not by fiat. The thesis is also honest about the simplifications it makes.

The soft spots, in proportion: the biggest one is not a hidden error—it's openly stated. The cloud mass Mc is treated as a free parameter (Sec 2.5), and scalar self-interactions are ignored (Sec 2.3). The stress-test note is right that this matters for the observational claim: for a given ALP model, a strong quartic coupling could trigger a bosenova and remove the cloud before the inspiral gets interesting. The thesis acknowledges this possibility but never maps the parameter region where the fiducial Mc/M = 0.01, alpha = 0.2 cloud actually survives. That leaves the "signature of the cloud" claims conditional—in the sense that if the cloud is there, the effects are large, but whether it is there depends on self-interactions.

Two smaller concerns: there is no code or data shipped, and no error bars on the numerical evaluations. The numerical convergence checks are mentioned but not documented in detail. For a thesis that's acceptable; for a paper claiming detectability, I'd want more.

Who is it for: graduate students and researchers entering the field who want the full derivation; also as a reference for the scaling laws. It deserves a serious referee if submitted as a review article, and the underlying papers already passed peer review. I'd recommend engaging with it, but cite the papers, not the thesis.

Referee Report

2 major / 7 minor

Summary. This PhD thesis studies the dynamics of an ultralight boson cloud ('gravitational atom') around a spinning black hole when a companion black hole inspirals through it. After reviewing the required background (Kerr black holes, gravitational waves, ultralight scalars, superradiance), the thesis develops a perturbative framework for the binary-cloud interaction and uses it to treat four effects: cloud-mediated dynamical capture, bound-state resonances, ionization into the continuum, and accretion onto the companion. The presentation is extended to eccentric and inclined orbits, and the backreaction of resonances on the binary is modeled through coupled Landau-Zener-type equations, leading to floating and sinking orbits and to preferred final values of eccentricity and inclination. The central quantitative claim, stated in the abstract and developed in Chapters 3-4, is that ionization power can exceed gravitational-wave luminosity during a substantial part of the inspiral, so that the cloud significantly modifies the merger time and leaves distinctive imprints in the binary parameters.

Significance. If its central claims hold, the work is significant: it provides a coherent, self-contained treatment of gravitational-atom binaries and identifies observable signatures that could be searched for with LISA, DECIGO, Einstein Telescope, and similar observatories. The main strengths are the analytic derivations from hydrogenic wavefunctions, Fermi's Golden Rule, Landau-Zener theory, and matched asymptotic expansions; the explicit scaling formulas in Sec. 4.2, which allow results to be extrapolated away from the fiducial parameters; and the transparent statement of assumptions at each step. The thesis is also honest in acknowledging its main limitations, in particular the treatment of the cloud mass as an independent parameter and the neglect of scalar self-interactions. Those limitations, however, are load-bearing for the observability claims, and the manuscript does not yet provide the model-space boundary needed to make the signatures falsifiable predictions for concrete ultralight-scalar models.

major comments (2)

[Secs. 2.3, 2.5, 3.4, 4.3] The abstract claims that ionization and resonances leave distinctive observational signatures, but the quantitative support for this claim assumes a cloud that survives intact with Mc/M treated as a free parameter. Section 2.3 states that self-interactions are 'ignored altogether', and Section 2.5 states that Mc is treated as an independent parameter rather than linked to the black-hole parameters, with only a passing remark that self-interactions can limit the cloud mass. For the ALP/QCD-axion models motivating the work, the quartic coupling in Eq. (2.3.4) is generically nonzero; strong self-interactions can trigger a bosenova that depletes the cloud and can also shift the bound-state spectrum entering the resonance conditions. The manuscript never computes the maximum stable Mc/M as a function of the decay constant f_a, nor does it map the region of scalar-model parameter space in which the fiducial cloud (Mc/M = 0.01, alpha = 0.2, used e.g. in Fig. 4.2) actually survives. Because P_ion/P_GW and the resonance strengths scale with the cloud mass and its survival, this is not a peripheral caveat: without this boundary, the signatures are conditional on an unquantified model assumption, and for a given ALP model the predicted effect can be absent simply because the cloud is gone. I ask the authors to either compute the bosenova-safe region for the parameter space considered, or to explicitly reframe the abstract and Chapter 7 claims as predictions conditional on a stated range of f_a and self-coupling.
[Sec. 5.1, Eqs. (5.1.2)-(5.1.3)] A central stated goal of the thesis is to extend the treatment of resonances and ionization to orbits with generic eccentricity and inclination. However, after deriving the separate eccentric and inclined expansions, Section 5.1 states: 'We do not explicitly compute η(g) in the general case'. Since the resonance strength Z, the Landau-Zener dynamics, and the resonance-breaking conditions studied in Sections 5.2-5.5 all depend quantitatively on η(g), the general simultaneous eccentric-and-inclined case is not actually evaluated. This missing step should be supplied, at least for a representative set of (epsilon, beta) values, or the scope of the 'generic orbit' claim should be explicitly restricted to the separately treated cases.

minor comments (7)

[Sec. 1] There is a typo in the introduction: 'photoeletric' should be 'photoelectric'.
[Sec. 2.1] In the discussion of Kerr black holes, 'astropyhsically' should be 'astrophysically'.
[Sec. 2.2.3] The word 'anormous' in the gravitational-wave detector overview should be 'enormous'.
[Sec. 4.6.3] The sentence 'ionization acts on inlined orbits' should read 'ionization acts on inclined orbits'.
[Fig. 4.8] The vertical axes of the ionization rate, power, and torque plots are labeled in arbitrary units; please provide the normalization factors in the caption or text so that the curves can be compared quantitatively with Figs. 3.5 and 4.5.
[Sec. 4.1] The claim that the dynamical-friction power Pdf reproduces Pion up to a universal O(1) factor depends on the choices rho = Mc|psi|^2, v = orbital velocity, and bmax = R*; the sensitivity of this comparison to the Coulomb cutoff bmax should be quantified.
[Abstract, Secs. 4.3 and 7] The statement that the predictions 'can be tested' with current and future interferometers is not yet tied to a detector-level calculation: I did not find signal-to-noise or waveform-mismatch estimates that show the predicted dephasing or parameter shifts are measurable. A quantitative detectability statement, even for one benchmark, would considerably strengthen the observational claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central predictions are derived from first-order perturbation theory, energy/momentum balance, and the hydrogenic spectrum; no fitted target quantity is renamed as a prediction.

full rationale

The thesis's central quantitative claims—ionization power Pion, the ionization rate, the Pion/Pgw ratio, resonance thresholds, and the preferred inclinations/eccentricities—are computed, not fitted. Section 3.4 obtains Pion via Fermi's Golden Rule (Eq. 3.4.5) from bound-to-continuum matrix elements of the Newtonian perturbation (Eq. 3.1.1), with the hydrogenic bound and continuum wavefunctions given in Sec. 2.5 and App. A. Section 4.4 re-derives the same result from a chirping-frequency Schrödinger equation (Eqs. 4.4.28–4.4.29), confirming the stationary-frequency result. The resonance backreaction equations (5.2.8)–(5.2.10) follow from conservation of energy and angular momentum coupled to the Landau–Zener system; no parameter is adjusted to match the claimed signatures. The thesis repeatedly cites the author's own prior papers [1–5], but those papers' derivations are reproduced in the thesis itself (Chapters 3–7), so the citations are not load-bearing in the sense of importing an unverified uniqueness theorem. The acknowledged modeling choices—treating Mc as an independent parameter (Sec. 2.5) and ignoring self-interactions (Sec. 2.3)—are assumptions that bound the validity of the predictions, not a circular reduction of the predictions to their own inputs. The skeptic concern that a bosenova could destroy the cloud is a physical robustness/assumption risk, not a circularity of the derivation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the prior superradiance and gravitational-atom formalism, the approximation of negligible self-interactions, the test-mass treatment of the companion, and standard quantum-mechanical tools. The only free parameters are physical system parameters such as Mc, alpha, and q, plus one arbitrary cutoff in a comparison calculation; no new particle, force, or entity is introduced.

free parameters (4)

cloud mass fraction Mc/M = 0.01 (fiducial; varied 0.001-0.1)
Treated as an independent parameter in Sec. 2.5; all ionization powers and resonance strengths scale linearly with Mc, so this choice directly sets the size of the predicted signals.
gravitational fine structure constant alpha = 0.2 (fiducial; also 0.1 in Fig. 3.3)
Chosen as input for numerical exploration, not fitted; however the results are only valid for alpha << 1, and the detectability claims use these values.
mass ratio q = 1e-3 (fiducial)
Chosen to match extreme and intermediate mass ratio inspirals; many scalings assume q << 1.
dynamical friction Coulomb cutoff bmax = R* (orbital separation)
In Sec. 4.1, bmax is chosen by hand to regulate a log divergence when comparing Pdf to Pion; the ratio is O(1) and not used for the main predictions.

assumptions (6)

domain assumption Rotational superradiance of a light scalar around a Kerr black hole creates a long-lived bound boson cloud with the stated growth rates.
Used as starting point in Secs. 2.4-2.5, following [24,27,130,131]; the thesis does not rederive the full superradiant instability from first principles.
domain assumption The scalar self-interactions are negligible, so the cloud is described by the linear Klein-Gordon and Schrodinger equation.
Stated in Sec. 2.3: '[w]e will ignore this altogether.' This is load-bearing for the hydrogenic level structure and for the absence of bosenova depletion.
domain assumption The companion is a test mass with q << 1 and the cloud's backreaction on the spacetime is O(Mc/M) and neglected.
Sec. 3.1 sets mass ratio q << 1 and treats the companion as a pointlike perturbation; Sec. 2.5 ignores the cloud's effect on the geometry.
standard math The nonrelativistic hydrogenic approximation (alpha << 1) with bound and continuum states of the Schrodinger-Coulomb problem is accurate at the order used.
Used in Secs. 2.5, 3.4, and Appendices A-B; the energy splittings include corrections up to O(alpha^5), but the wavefunctions are hydrogenic to leading order.
standard math Fermi's Golden Rule and the Markov (Weisskopf-Wigner) approximation correctly capture bound-to-continuum ionization rates.
Central to Secs. 3.4 and 4.4; the thesis argues that the chirp transient is subleading after a timescale gamma^{-1/2}.
standard math The Landau-Zener formula governs bound-bound resonances, with an extended numerical threshold for backreaction.
Used in Secs. 3.3 and 5.3; the critical value ZB = 0.1686 is obtained by numerical exploration, not by an analytic proof.

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Cite this review

Pith. "Pith review of Gravitational Atoms and Black Hole Binaries." pith.science (2026). https://pith.science/paper/TR3TXRAU

@misc{pith2026241212526,
  author       = {Pith},
  title        = {Pith review of: Gravitational Atoms and Black Hole Binaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TR3TXRAU}},
  note         = {Machine review of arXiv:2412.12526}
}

read the original abstract

Several models of physics beyond the Standard Model predict the existence of new ultralight bosons. This thesis investigates a way to discover such particles through observations of gravitational waves from binary black holes. This is possible through black hole superradiance, which spontaneously creates a "boson cloud" around a rapidly spinning black hole. The system is also known as a gravitational atom, due to its similarities with the hydrogen atom. The thesis focuses on a scenario where a gravitational atom is orbited by a binary companion. The goal is to characterize the dynamics of the system and identify the signatures left by the boson cloud on the gravitational waves emitted by the binary. The predictions can be tested with current and future interferometers, such as LISA, LIGO, DECIGO, Einstein Telescope and TianQin. First, I demonstrate that the cloud catalyzes the binary formation by increasing the dynamical capture cross section. I then introduce and study the ionization of the cloud, wherein the perturbation from the binary unbinds the bosons, analogous to the photoelectric effect in atomic physics. After that, I examine the accretion of the cloud on the companion black hole. To achieve realistic and complete results, I proceed to extend the treatment of ionization, as well as of the orbital resonances discussed in earlier works, to orbits with generic inclination and eccentricity. This allows to study the entire history of the system, from formation to merger. The most distinctive observational signatures of the cloud are found to be the orbital energy lost through ionization and the preference for specific inclinations and eccentricities induced by the orbital resonances.

Figures

Figures reproduced from arXiv: 2412.12526 by the authors.

**Figure 1.1.** Illustration of a gravitational atom in a binary system. As the two black [PITH_FULL_IMAGE:figures/full_fig_p014_1_1.png] view at source ↗

**Figure 2.1.** Sensitivity curves of current (solid lines) and future (dashed lines) [PITH_FULL_IMAGE:figures/full_fig_p026_2_1.png] view at source ↗

**Figure 2.2.** Regge plane of black holes. The blue shaded area corresponds to the [PITH_FULL_IMAGE:figures/full_fig_p034_2_2.png] view at source ↗

Figures from the paper (39 more)

**Figure 2.3.** Figure 2.3: Equatorial cross section of a gravitational atom in the [PITH_FULL_IMAGE:figures/full_fig_p038_2_3.png]

**Figure 2.4.** Figure 2.4: Illustration of the spectrum of bound and unbound states of the gravita [PITH_FULL_IMAGE:figures/full_fig_p040_2_4.png]

**Figure 3.1.** Figure 3.1: Schematic illustration of the binary system. The primary object of the [PITH_FULL_IMAGE:figures/full_fig_p043_3_1.png]

**Figure 3.2.** Figure 3.2: Energy lost to the cloud (3.2.8) as function of the distance of closest approach Rp, for α = 0.2, q = 10−3 , Mc = 0.01M and a cloud in the |211⟩ state. The i subscript denotes each of the two contributions to (3.2.8), so that Elost = P i Ei lost. Thick (thin) lines r…

**Figure 3.3.** Figure 3.3: Capture cross section σtot, including the energy lost to both the cloud and GWs, normalized by capture cross section (3.2.1) due to GWs only. The cross section is shown as a function of the relative asymptotic velocity between the two objects, v. Thick lines are comp…

**Figure 3.4.** Figure 3.4: Numerical solution of (3.3.7). An adiabatic transition with full transfer from |a⟩ to |b⟩ is observed for Z = 25 (left panel), while a partial transfer is observed for for Z = 0.25 (right panel). In both cases, the final populations at t → +∞ asymptote those given in…

**Figure 3.5.** Figure 3.5: Ionization power (3.4.5) as function of the orbital separation R∗, for α = 0.2, q = 10−3 , Mc = 0.01M and a cloud in the |211⟩ state. The top panel shows Pion in units where M = 1. The bottom panel shows the ratio of Pion to Pgw, the power lost due to GW emission (2.…

**Figure 3.6.** Figure 3.6: Cartoon illustrating the accretion of the boson cloud by the companion [PITH_FULL_IMAGE:figures/full_fig_p055_3_6.png]

**Figure 3.7.** Figure 3.7: Schematic illustration of the near-field and far-field expansions, where [PITH_FULL_IMAGE:figures/full_fig_p057_3_7.png]

**Figure 3.8.** Figure 3.8: Mass accretion rate of a Schwarzschild black hole computed analytically— [PITH_FULL_IMAGE:figures/full_fig_p059_3_8.png]

**Figure 4.1.** Figure 4.1: Comparison of Pdf (divided by 4 for clarity) with Pion, for clouds in the states |311⟩, |322⟩ and |422⟩. All the parameters and the units are the same as in [PITH_FULL_IMAGE:figures/full_fig_p067_4_1.png]

**Figure 4.2.** Figure 4.2: Evolution of the separation R∗, for M = 104M⊙ and α = 0.2, with initial values of R∗ = 400M, q = 10−3 and Mc/M = 0.01 in a |211⟩ state. Shown are the results for both co-rotating and counter-rotating orbits. The vacuum system, where no cloud is present, is shown for …

**Figure 4.3.** Figure 4.3: Fractional changes of the mass of the cloud [PITH_FULL_IMAGE:figures/full_fig_p070_4_3.png]

**Figure 4.4.** Figure 4.4: The imaginary part of the induced energy [PITH_FULL_IMAGE:figures/full_fig_p073_4_4.png]

**Figure 4.5.** Figure 4.5: Ionization power (4.5.6) for different values of the eccentricity ε, as function of the semi-major axis a. The values are normalized by Pgw, the average power emitted in gravitational waves on a correspondingly eccentric orbit, and are computed for α = 0.2, q = 10−3…

**Figure 4.6.** Figure 4.6: Numerical solutions to (4.5.12), for various different initial values of the semi-major axis and the eccentricity. The top panel neglects Pgw and τgw, while the bottom panel shows the solution to the complete equation. The values of the parameters and the orientation…

**Figure 4.7.** Figure 4.7: Diagram of the coordinates used to describe inclined orbits. The orbital [PITH_FULL_IMAGE:figures/full_fig_p085_4_7.png]

**Figure 4.8.** Figure 4.8: Instantaneous ionization rate (top), power (middle) and torque along z (bottom) for a cloud in the |211⟩ state, as function of the binary separation, for different values of the orbital inclination β. The y axes are reported in arbitrary units (a.u.), while the x axi…

**Figure 4.9.** Figure 4.9: Variation of the inclination angle ∆β ≡ β − β0, as function of the orbital separation R∗, for different values of the initial inclination β0. The curves represent the evolution of ∆β, from right to left, over the course of an inspiral. Solid lines are obtained by dir…

**Figure 5.1.** Figure 5.1: Numerical solution of the nonlinear system ( [PITH_FULL_IMAGE:figures/full_fig_p099_5_1.png]

**Figure 5.2.** Figure 5.2: Floating (left panel) and sinking (right panel) resonances on eccentric orbits, with ∆m/g = 1. We display the value of the frequency ω, the eccentricity ε and the populations |ca| 2 and |cb| 2 as function of τ , obtained by solving equations (3.3.7), (5.2.8) and (5.2…

**Figure 5.3.** Figure 5.3: Same resonances as in Figure [PITH_FULL_IMAGE:figures/full_fig_p102_5_3.png]

**Figure 5.4.** Figure 5.4: Flow in the eccentricity-inclination plane ( [PITH_FULL_IMAGE:figures/full_fig_p102_5_4.png]

**Figure 5.5.** Figure 5.5: Numerical solution of equations (3.3.7), (5.2.8), (5.2.9) and (5.2.10) with parameters Z = 0.001, B = 1000, D = 4/3 and g = ∆m. For simplicity we ignore that in realistic cases Z depends on the eccentricity, and we keep it constant instead. The system is initialized …

**Figure 5.6.** Figure 5.6: Numerical solution of equations (3.3.7) and (5.2.8) with (initial) parameters Z = 0.001 and B = 1000. A Z-breaking occurs when Z is slowly reduced over time, with the resonance ending when (5.3.18) is satisfied (left panel). A Γ-breaking is observed when Z is kept f…

**Figure 5.7.** Figure 5.7: Position of a few selected Bohr resonances, compared to [PITH_FULL_IMAGE:figures/full_fig_p111_5_7.png]

**Figure 6.1.** Figure 6.1: Floating timescale ∆tfloat (solid lines), compared to the decay timescale tdecay (dashed lines) of the final state, for some selected resonances. We use benchmark parameters and determine the decay rate independently through Leaver’s continued fraction method [27, …

**Figure 6.2.** Figure 6.2: Illustration of the possible outcomes of the resonant history of the cloud [PITH_FULL_IMAGE:figures/full_fig_p118_6_2.png]

**Figure 6.3.** Figure 6.3: Mass of the cloud Mbr c at resonance Γ-breaking, as function of α and β, for the two hyperfine resonances from the initial state |211⟩. The mass of the cloud decreases during the resonance from its initial value Mc, and the resonance breaks when the value Mbr c is re…

**Figure 6.4.** Figure 6.4: Function F(α, Mc) appearing in equation (6.2.8), which defines the angular interval δ2 around a counter-rotating orbit where the resonance |211⟩ → |200⟩ is not adiabatic. whose angular dependence is determined through (5.5.6) as usual. This resonance, however, has a…

**Figure 6.5.** Figure 6.5: Strongest sinking Bohr resonances on a counter-rotating orbit for a clound [PITH_FULL_IMAGE:figures/full_fig_p122_6_5.png]

**Figure 6.6.** Figure 6.6: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p124_6_6.png]

**Figure 7.1.** Figure 7.1: Same as in Figure [PITH_FULL_IMAGE:figures/full_fig_p128_7_1.png]

**Figure 7.2.** Figure 7.2: Fractional change of the mass M∗ of the secondary and the mass Mc of the cloud. The parameters are the same as [PITH_FULL_IMAGE:figures/full_fig_p128_7_2.png]

**Figure 7.3.** Figure 7.3: Evolution of the GW frequency as a function of the remaining time to [PITH_FULL_IMAGE:figures/full_fig_p129_7_3.png]

**Figure 7.4.** Figure 7.4: Evolution of the (inverse) frequency fgw for M = 104M⊙ and α = 0.04, 0.08, . . . , 0.28, with initial q = 10−3 and Mc/M = 0.01 in a |211⟩ state. The axes are rescaled according to Eq. (7.1.8), with ˆα ≡ α/0.2. The curves have been horizontally shifted to match at t =…

**Figure 7.5.** Figure 7.5: Example values of the fixed point ¯ε depending on ∆m/g. Numbers can be found by solving equations (5.2.9) and (5.2.10) on a floating orbit. time of3 ∆tfloat = 5.8 yrs M 104M⊙ q 10−3 −2 α 0.2 −3 . (7.1.11) Although the cloud’s mass is continuously reduced by i…

**Figure 7.6.** Figure 7.6: Examples of backreaction on the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p133_7_6.png]

**Figure 7.7.** Figure 7.7: The shaded regions show the possible values of eccentricity [PITH_FULL_IMAGE:figures/full_fig_p134_7_7.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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