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Detection of Intermediate-Mass Ratio Inspirals in Globular Clusters: Revealing the Brownian Motion with Gravitational Waves

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Brownian motion of IMRIs in globular clusters is imprinted on gravitational-wave phase strongly enough to be distinguishable from models that ignore it.

desk verdict A useful first pass at forecasting Brownian-motion phase shifts in IMRI detections, but the encounter model systematically overstates the effect and the abstract overclaims. read the letter →

arxiv 2501.13466 v2 pith:L66YITUT submitted 2025-01-23 astro-ph.HE gr-qc

Alejandro Torres-Orjuela , Verónica Vázquez-Aceves , Tian-Xiao Wang This is my paper · ORCID
classification astro-ph.HEgr-qc
keywords gravitationalwavesintermediate-massratioinspiralsblackholesglobularclustersBrownianmotionDopplerphaseshiftaberrationalwaveastronomy
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

An intermediate-mass-ratio inspiral (IMRI)—a stellar-mass compact object spiraling into a black hole of roughly $10^2$–$10^5\,M_\odot$—sitting in a globular cluster is constantly jostled by weak encounters with passing stars. This paper argues that the resulting Brownian motion of the IMRI's center of mass produces Doppler and aberrational phase shifts in the emitted gravitational waves that are large enough to see. Using 100 realistic systems drawn from cluster simulations, it finds that fewer than 10 percent would be bright enough to detect with TianQin, LISA, or AION, but for all of those detected sources the signal-to-noise ratio is high enough to tell a waveform that includes the jostling from one that ignores it, with matches mostly below 0.8. A sympathetic reader would care because the result turns an environmental nuisance into a measurable signal: if correct, future space-based observatories can both detect IMRIs and probe the stellar environment that stirred them, and waveform models must include Brownian motion to avoid losing the signal.

What carries the argument

The central mechanism is the Brownian motion of the IMRI's center of mass, modeled as a sequence of weak encounters with field stars in a Plummer-model cluster. The load-bearing pieces are the 1% binding-energy threshold that fixes a minimum encounter distance, the resulting upper-limit acceleration $a(r_*) \lesssim N c^2/(800 R_h d)\cdot r_*^2/(r_*^2+9/16)^{3/2}$, the lower-limit encounter duration from the escape-velocity requirement, and the encounter rate from the free-path length. These feed the Doppler phase shift $\Phi_{\rm Dop,n}(t)=\int dt' \beta(t') \cos\theta\, \omega_n(t')$ and aberrational shift $\Phi_{\rm Abe}(t)$, whose accumulated difference from an unaccelerated waveform is quantified by the noise-weighted match $\mathcal{M}$ and the SNR-based distinguishability criterion. The paper uses these to assert that the effect is detectable for all sources with SNR above threshold.

What would settle it

A Monte Carlo scattering calculation with the same 100 simulated IMRI systems that draws impact parameters, encounter durations, and acceleration directions from the cluster's actual stellar distribution, rather than fixing every encounter at maximum acceleration and minimum duration, would settle it: if the recomputed match stays above about 0.8 for most detectable sources at their nominal SNRs, the central claim is refuted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that weak stellar encounters, which do not alter the inspiral's orbital parameters, accelerate the IMRI's center of mass and imprint a time-dependent phase shift—a Doppler term from the line-of-sight velocity change plus an aberrational term from the transverse velocity change—on each harmonic of the gravitational wave. Over a five-year observation the accumulated phase shift can exceed 150 radians, far larger than the analytic estimate that assumes the shift is just the total velocity change divided by $c$. Comparing waveforms with and without this effect, the paper finds matches mostly below 0.8 and, using the criterion that two waveforms are distinguishable when $\rho > \sqrt{D/[2(1-\mathcal{M})]}$ with $D=15$ parameters, concludes that every source above the detection threshold can be identified as having undergone Brownian motion, with one TianQin source as the only exception. The paper claims this makes the Brownian motion of IMRIs a detectable environmental signature rather than a negligible correction.

Load-bearing premise

The load-bearing premise is that the extremal encounter model is representative: with a 1% binding-energy threshold, each encounter is assigned the maximum acceleration and minimum duration, treated as constant and direction-alternating, while the inspiral's orbital parameters stay fixed, so if real encounters are weaker, longer, or differently oriented, the predicted phase shifts and the claimed detectability would shrink.

Editorial extensions

If this is right

  • Only a small fraction—under 10 percent of the simulated IMRI population—will reach SNR above 15 in any single detector, and detectable sources sit within roughly the Local Group; LISA finds the most, then TianQin, then AION.
  • For all sources above threshold, and even some with SNR between 1 and 15, the Brownian-motion phase shift is large enough that the wrong waveform without encounters is distinguishable at the source's own SNR.
  • Gravitational-wave templates for IMRIs formed in dense clusters must include center-of-mass acceleration from weak encounters, or the mismatch will cause loss of signal.
  • The random, time-varying character of the induced phase shift lets it be separated from smoother environmental effects such as gas drag, and joint detection by two or three detectors modestly improves the SNR and the distinguishability.

Reading between the lines

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

  • Beyond the paper: the strong correlation the paper finds between acceleration, encounter rate, and encounter duration suggests a single well-measured IMRI phase-shift pattern could constrain the local stellar density and encounter rate, and a handful of detections could map central densities across globular clusters.
  • Beyond the paper: the same Doppler-aberration mechanism should apply to lighter stellar-mass black hole binaries and to extreme-mass-ratio inspirals in other dense environments, so the match-based test described here could be rerun for those source classes.
  • Beyond the paper: replacing the constant, extremal-acceleration encounter model with a realistic distribution of impact parameters would test whether the predicted match values are robust; this is the most direct way to check the paper's detectability estimate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper models the Brownian motion of intermediate-mass ratio inspirals (IMRIs) in globular clusters due to weak encounters with field stars, using a Plummer cluster model and 100 IMRI systems from MOCCA Survey I. It computes signal-to-noise ratios for TianQin, LISA, and AION, injects Doppler and aberrational phase shifts from the Brownian motion into the waveforms, and compares the resulting signals with a model that ignores the effect through the match and a Cutler-Vallisneri distinguishability criterion. The central claims are that fewer than 10% of the considered IMRIs are detectable, but that for detected sources the Brownian-motion phase shift is generally distinguishable, with matches mostly below 0.8, implying that future waveform models should include this environmental effect.

Significance. If the quantitative results survive a more careful treatment, this would be an important and timely demonstration that future mHz and dHz detectors can use gravitational waves to probe the dynamical environment of IMRIs rather than merely detect them. The use of a realistic MOCCA sample and a transparent analytic encounter model are strengths, and the central inference is not circular: the encounter acceleration, rate, and duration are derived from the cluster model rather than fitted to the match results. However, the quantitative prediction rests on a highly stylized encounter prescription, and the paper's own numbers contain an internal contradiction with the abstract's blanket distinguishability claim. A self-consistent treatment of the encounter distribution is needed before the predicted match distribution can be trusted.

major comments (4)
  1. [Sec. 2.2-2.3, Eqs. (5), (10), (12)] The encounter injection systematically overestimates the accumulated phase shift. Eq. (5) is a mean-free-path estimate with no impact-parameter cutoff, counting all stars crossed, while the injected acceleration is the maximum allowed value at the 1%-binding-energy impact parameter alpha_min of Eq. (9). Assigning a = a_max and Delta_t = Delta_t_min to every encounter makes each encounter change the velocity by v_esc, since the product of Eqs. (10) and (12) equals the escape velocity in Eq. (11). The rate of encounters with b less than or similar to alpha_min is far smaller than Gamma = 1/t_enc, so the model replaces a distribution of frequent weak kicks with repeated maximal kicks. This is not a conservative two-sided bound: the Doppler phase shift in Eq. (21) is an integral of the accumulated line-of-sight velocity, so both the number of kicks and their size enter multiplicatively. I recommend recomputing Table 3 with encounters drawn from the actual impact-parameter distribution, or at least with a rate restricted to b <= alpha_min, before trusting the <0.8 match result.
  2. [Sec. 2.3, Eqs. (7)-(12)] The 'weak encounter' condition is internally inconsistent. Setting Delta_Ekin <= 0.01 E_bind with Eq. (7) implies a velocity kick Delta_v = c / sqrt(200 d), which for d ~ 10^3 is several hundred km/s, far larger than the escape velocity in Eq. (11) for the clusters considered; the model then caps the kick at v_esc through the duration in Eq. (12). Thus the injected signal is not the weak-encounter limit described in the text, and the 1% threshold is not what actually controls the phase shift. A self-consistent treatment should derive acceleration and duration from the same encounter geometry without imposing v_esc by hand.
  3. [Sec. 5.2, Table 3, and Sec. 6] The blanket claim that all detected sources are distinguishable is contradicted by the paper's own numbers. Source MCH has SNRs 16.7/37.2/38.4 and matches 1.0/0.941/0.999 in AION/LISA/TianQin; for TianQin, Eq. (16) with D = 15 gives rho_required about 87, well above 38.4, and for AION the match is exactly unity. The text already notes 'the only exception being one of the sources detected by TianQin,' so the abstract's statement 'for all sources detected, the SNR is high enough to discern the Brownian motion' and the conclusion's 'all detectable sources have a mismatch big enough' should be revised or qualified.
  4. [Sec. 2.3 and Sec. 5] The results contain no propagated uncertainty or sensitivity analysis for the model choices that set the effect size: the 1% binding-energy threshold and the constant, direction-alternating acceleration profile. A scan over the threshold (e.g., 0.1% to 10%) and a test with random acceleration orientations would show whether the match distribution in Table 3 is robust or is a consequence of these choices. As written, the required-SNR values in Figs. 5 and 6 inherit these unquantified systematics and should not be interpreted as precise predictions.
minor comments (4)
  1. [Eq. (22)] The definition mu = sin^{-1}(delta) is dimensionally inconsistent because delta is an angle; please check whether 1/sin(delta) or another quantity is intended.
  2. [Eq. (5)] The typeset formula appears to be missing a division by N in the denominator; combining Eqs. (3) and (4) gives t_enc proportional to R_h^{3/2} / (sqrt(G M) N).
  3. [Sec. 4 and Sec. 5.1] The paper alternates between a five-year observation time and LISA's three-year operation time; it should be stated explicitly which observation time is used in the match computations.
  4. [Sec. 5.4] Since the phase shift can exceed 150 rad, the linear-in-delta_Phi approximation underlying Eq. (17) is violated; the qualitative conclusion for very low matches is safe, but the numerical required-SNR values in Figs. 5-8 should be labeled as order-of-magnitude estimates.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the detectability estimate is a forward calculation from independent cluster and source inputs.

full rationale

The paper's central claim is a forward-model detectability estimate, not a fit disguised as a prediction. The acceleration (Eq. 10), encounter duration (Eq. 12), and encounter rate (Eq. 5) are derived from a Plummer-model cluster profile and from MOCCA simulation data for 100 IMRI systems; these inputs are external to the match calculation and are not adjusted to produce the low match values. The SNR and match are then computed by injecting the resulting phase shift into a waveform and comparing it with a waveform without the effect, which is a standard injection-style calculation. The phase-shift formalism (Doppler and aberrational terms, Eqs. 20-22) is cited from the authors' earlier papers, but it is a published, independently usable relativistic result, and it is used as a modeling premise rather than as evidence for the conclusion; the conclusion that Brownian motion is distinguishable follows from the numerical size of the injected phase shift and the SNR criterion. The paper explicitly acknowledges modeling simplifications (constant acceleration, taking the maximum acceleration and minimum duration, assuming all encounters have the same magnitude), which are limitations on robustness but are not circular inputs. No equation in the paper is defined in terms of the target result, and no fitted parameter is renamed as a prediction. Self-citations appear, but they are not load-bearing in the sense of forcing the paper's central claim by definition.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central forecast depends on the Plummer encounter model, the weak-encounter assumption, the phase-shift formalism from prior work, and a hand-set acceleration profile. No unknown constants are fitted to the match values, but several order-of-magnitude choices, such as the 1% binding-energy threshold and the maximum-acceleration/minimum-duration profile, shape the claimed detectability.

free parameters (2)
  • Encounter binding-energy threshold = 0.01 (DeltaEkin <= 0.01 Ebind)
    Chosen by hand in Sec. 2.3; sets the minimum encounter distance and therefore the maximum acceleration in Eq. (10) and the duration in Eq. (12). The whole phase-shift and match analysis scales with this choice.
  • Illustrative semi-major axis factor d = 1000
    Assumed for the representative cluster curves in Figs. 1 and 2 and the Sec. 4 order-of-magnitude SNR estimate; the main 100-source analysis uses actual d from the MOCCA initial semi-major axes.
assumptions (8)
  • domain assumption The host cluster is well described by a Plummer density profile, Eq. (1).
    This profile is used to derive the free path, encounter time, acceleration, and duration in Eqs. (3)-(12); real MOCCA clusters have more complex structure.
  • standard math The IMRI speed follows from the virial theorem for the mass enclosed in a Plummer sphere, Eq. (4).
    Standard dynamical estimate; ignores velocity anisotropy and the IMBH's own potential, which could change encounter rates.
  • domain assumption Each mean-free path sF defined by integrating the stellar number density to one encounter gives the encounter rate.
    Eqs. (3)-(5) treat encounters as purely radial and ignore the impact-parameter distribution, an idealization the paper acknowledges in Sec. 2.2.
  • domain assumption Weak encounters do not change the IMRI's orbital parameters, so only center-of-mass acceleration matters.
    Sec. 2.3 excludes disruptive encounters via a 1% binding-energy bound, but real encounters include a spectrum of perturbations to the binary elements.
  • standard math The Doppler and aberrational phase-shift formulas, Eqs. (21)-(22), describe the gravitational-wave phase of an accelerated source.
    Taken from the authors' prior works; assumed valid for v much less than c and high inclination.
  • domain assumption The Barack and Cutler (2004) EMRI waveform, with leading-order eccentricity and mode truncation, is adequate for these IMRI signals.
    Used for all SNR and match computations; the paper notes the leading-order eccentricity approximation is imperfect for eccentricity close to 1.
  • standard math The Cutler and Vallisneri (2007) distinguishability criterion with D=15 parameters correctly separates the two waveform models.
    This criterion sets the required SNR for a given mismatch; D=15 is an assumption about which parameters are unknown.
  • ad hoc to paper The encounter can be represented as a constant acceleration equal to the upper bound in Eq. (10), lasting the lower bound in Eq. (12), and alternating direction.
    This is the key hand-made profile that generates the injected phase shifts; it is not derived from a full encounter simulation and no alternative profiles are tested.

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Cite this review

Pith. "Pith review of Detection of Intermediate-Mass Ratio Inspirals in Globular Clusters: Revealing the Brownian Motion with Gravitational Waves." pith.science (2026). https://pith.science/paper/L66YITUT

@misc{pith2026250113466,
  author       = {Pith},
  title        = {Pith review of: Detection of Intermediate-Mass Ratio Inspirals in Globular Clusters: Revealing the Brownian Motion with Gravitational Waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L66YITUT}},
  note         = {Machine review of arXiv:2501.13466}
}
abstract

Intermediate-mass ratio inspirals (IMRIs) formed by stellar-mass compact objects orbiting intermediate-mass black holes will be detected by future gravitational wave (GW) observatories like TianQin, LISA, and AION. We study a set of 100 IMRI systems in globular clusters obtained from MOCCA simulations to estimate their detectability. Furthermore, we model the Brownian motion of the IMRIs induced by weak interactions with the surrounding field of stars and include its effect on the GW's phase through Doppler and aberrational phase shift. We find that a small fraction of IMRIs ($<10\,\%$) will have signal-to-noise ratios (SNR) high enough to be detected by TianQin, LISA, and AION. However, for all sources detected, the SNR is high enough to discern the Brownian motion of the IMRI. More precisely, we find that the match between the signal containing the effect of the Brownian motion and a waveform model without this effect is mostly low ($<0.8$). These results highlight the importance of including the interaction of IMRIs with the surrounding field of stars to obtain proper detection, but also show the possibility of studying the environment of the source using GWs.

Figures

Figures reproduced from arXiv: 2501.13466 by the authors.

Figure 1
Figure 1. The time between two encounters ten (solid line) and the velocity of the IMRI (dotted line) as functions of the distance from the center of the star cluster r∗. We show these functions for a ‘big cluster’ (red) and a ’small cluster’ (blue). See the text for more details. spherical coordinates from zero to r∗. Combining Eq. (3) and Eq. (4), we then find for the time between two en￾counters tenc(r∗) = 16 27 R 3/2 √ h … view at source ↗
Figure 2
Figure 2. The acceleration a of an IMRI due to its Brow￾nian motion (solid line) and the duration of an encounter ∆t (dotted line) as functions of the distance from the center of the star cluster r∗. a and ∆t are shown for a ‘big cluster’ (red) and a ’small cluster’ (blue). See the text for more de￾tails. and assuming that vint is smaller than the escape veloc￾ity vesc(r∗) = r 2GM Rh [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The relevant parameters for the encounters of an IMRI in the different clusters considered. The abscissa shows the estimated acceleration during an encounter a, the ordinate, the estimated ratio of encounters per year Γ, and the color encodes the duration of an encounter ∆t. tions, short encounters, and frequent encounters, while low accelerations correlate with long infrequent encoun￾ters. The strong correlation be… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The SNR of the IMRIs in AION (yellow dia￾monds), LISA (blue triangles), and TianQin (red circles) as a function of the source’s luminosity distance DL, where we only include sources with a SNR of at least 1. The black dotted line corresponds to the threshold SNR of 15 …
Figure 5
Figure 5. Figure 5: The match (upper plot) and the difference be￾tween the SNR of the source and the SNR required for the detection of the acceleration (lower plot) as a function of the acceleration a. We show the values for AION (yellow di￾amonds), LISA (blue triangles), and TianQin (red…
Figure 6
Figure 6. Figure 6: The match (upper plot) and the difference between the SNR of the source and the SNR required for the detection of the encounters (lower plot) as a function of the ratio of encounters Γ. We use the same convention as in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: The SNR of the IMRIs for joint detection be￾tween AION and LISA (green diamonds), AION and Tian￾Qin (orange triangles), LISA and TianQin (purple circles), and all three detectors (black crosses). We show the SNR de￾pending on the luminosity distance of the source DL, w…
Figure 8
Figure 8. Figure 8: shows the detectability of the acceleration from joint detection. Similar to the case of the SNR, joint detection allows for better detectability – mainly due to the higher total SNR – but the difference is not very prominent. Most remarkably, we find that in some case…
Figure 9
Figure 9. Figure 9: The phase shift δΦ = Φa − Φ0 between the signal affected by the acceleration of the source (Φa) and a waveform ignoring the effect of the encounters (Φ0) for some exemplary sources. SNRs of just a few tens is enough to differentiate be￾tween the signal and the model; i…

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Works this paper leans on

80 extracted references · 15 canonical work pages

  1. [1]

    2024, arXiv e-prints, arXiv:2412.14960, doi: 10.48550/arXiv.2412.14960

    Abdalla, A., Abe, M., Abend, S., et al. 2024, arXiv e-prints, arXiv:2412.14960, doi: 10.48550/arXiv.2412.14960

    arXiv doi:10.48550/arxiv.2412.14960 2024

  2. [2]

    2024, A VS Quantum Science, 6, 024701, doi: 10.1116/5.0185291 Abou El-Neaj, Y., Alpigiani, C., Amairi-Pyka, S., et al

    Abend, S., Allard, B., Alonso, I., et al. 2024, A VS Quantum Science, 6, 024701, doi: 10.1116/5.0185291 Abou El-Neaj, Y., Alpigiani, C., Amairi-Pyka, S., et al. 2020, EPJ Quantum Technology, 7, 6, doi: 10.1140/epjqt/s40507-020-0080-0

    doi:10.1116/5.0185291 2024

  3. [3]

    2023, Living Reviews in Relativity, 26, 2, doi: 10.1007/s41114-022-00041-y

    Amaro-Seoane, P., Andrews, J., Arca Sedda, M., et al. 2023, Living Reviews in Relativity, 26, 2, doi: 10.1007/s41114-022-00041-y

    doi:10.1007/s41114-022-00041-y 2023

  4. [4]

    2014, ApJ, 781, 45, doi: 10.1088/0004-637X/781/1/45

    Antonini, F., Murray, N., & Mikkola, S. 2014, ApJ, 781, 45, doi: 10.1088/0004-637X/781/1/45

    doi:10.1088/0004-637x/781/1/45 2014

  5. [5]

    L., Petrovich, C., & Fischer, C

    Antonini, F., Rodriguez, C. L., Petrovich, C., & Fischer, C. L. 2018, MNRAS, 480, L58, doi: 10.1093/mnrasl/sly126 Arca Sedda, M., Amaro Seoane, P., & Chen, X. 2021, A&A, 652, A54, doi: 10.1051/0004-6361/202037785

    doi:10.1093/mnrasl/sly126 2018

  6. [6]

    2016, Monthly Notices of the Royal Astronomical Society: Letters, 464, L36–L40, doi: 10.1093/mnrasl/slw177

    Askar, A., Szkudlarek, M., Gondek-Rosi´ nska, D., Giersz, M., & Bulik, T. 2016, Monthly Notices of the Royal Astronomical Society: Letters, 464, L36–L40, doi: 10.1093/mnrasl/slw177

    doi:10.1093/mnrasl/slw177 2016

  7. [7]

    2020, JCAP, 05, 011, doi: 10.1088/1475-7516/2020/05/011

    Badurina, L., et al. 2020, JCAP, 05, 011, doi: 10.1088/1475-7516/2020/05/011

    doi:10.1088/1475-7516/2020/05/011 2020

  8. [8]

    F., Reines, A

    Baldassare, V. F., Reines, A. E., Gallo, E., & Greene, J. E. 2015, ApJL, 809, L14, doi: 10.1088/2041-8205/809/1/L14

    doi:10.1088/2041-8205/809/1/l14 2015

Show all 80 references
  1. [9]

    2004, Phys

    Barack, L., & Cutler, C. 2004, Phys. Rev. D, 69, 082005, doi: 10.1103/PhysRevD.69.082005

    2004 doi

  2. [10]

    2008, Galactic Dynamics: Second Edition (Princeton University Press)

    Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press)

    2008

  3. [11]

    2016, MNRAS, 461, 1023, doi: 10.1093/mnras/stw1372

    Mapelli, M. 2016, MNRAS, 461, 1023, doi: 10.1093/mnras/stw1372

    2016 doi

  4. [12]

    2002, PhRvL, 88, 121103, doi: 10.1103/PhysRevLett.88.121103

    Chatterjee, P., Hernquist, L., & Loeb, A. 2002, PhRvL, 88, 121103, doi: 10.1103/PhysRevLett.88.121103

    2002 doi

  5. [13]

    2024, arXiv e-prints, arXiv:2402.07571, doi: 10.48550/arXiv.2402.07571

    Colpi, M., Danzmann, K., Hewitson, M., et al. 2024, arXiv e-prints, arXiv:2402.07571, doi: 10.48550/arXiv.2402.07571

    2024 doi arXiv

  6. [14]

    2007, PhRvD, 76, 104018, doi: 10.1103/PhysRevD.76.104018 Dall’Amico, M., Mapelli, M., Torniamenti, S., & Arca

    Cutler, C., & Vallisneri, M. 2007, PhRvD, 76, 104018, doi: 10.1103/PhysRevD.76.104018 Dall’Amico, M., Mapelli, M., Torniamenti, S., & Arca

    2007 doi

  7. [15]

    2024, A&A, 683, A186, doi: 10.1051/0004-6361/202348745

    Sedda, M. 2024, A&A, 683, A186, doi: 10.1051/0004-6361/202348745

    2024 doi

  8. [16]

    B., Ferdman, R

    Demorest, P. B., Ferdman, R. D., Gonzalez, M. E., et al. 2013, ApJ, 762, 94, doi: 10.1088/0004-637X/762/2/94

    2013 doi

  9. [17]

    N., Lentati, L., et al

    Desvignes, G., Caballero, R. N., Lentati, L., et al. 2016, MNRAS, 458, 3341, doi: 10.1093/mnras/stw483 Di Cintio, P., Ciotti, L., & Nipoti, C. 2020, in IAU

    2016 doi

  10. [18]

    351, Star Clusters: From the Milky Way to the Early Universe, ed

    Symposium, Vol. 351, Star Clusters: From the Milky Way to the Early Universe, ed. A. Bragaglia, M. Davies, A. Sills, & E. Vesperini, 93–96, doi: 10.1017/S1743921319007221

    doi

  11. [19]

    1905, Annalen der Physik, 322, 549, doi: 10.1002/andp.19053220806

    Einstein, A. 1905, Annalen der Physik, 322, 549, doi: 10.1002/andp.19053220806

    1905 doi

  12. [20]

    Finn, L. S. 1992, PhRvD, 46, 5236, doi: 10.1103/PhysRevD.46.5236

    1992 doi

  13. [21]

    Rasio, F. A. 2004, MNRAS, 352, 1, doi: 10.1111/j.1365-2966.2004.07914.x

    2004

  14. [22]

    2022, MNRAS, 517, 1339, doi: 10.1093/mnras/stac2711

    Mayer, L. 2022, MNRAS, 517, 1339, doi: 10.1093/mnras/stac2711

    2022 doi

  15. [23]

    M., & Ho, L

    Gebhardt, K., Rich, R. M., & Ho, L. C. 2002, apjl, 578, L41, doi: 10.1086/342980

    2002 doi

  16. [25]

    2001b, in Astronomical Society of the Pacific Conference Series, Vol

    Giersz, M. 2001b, in Astronomical Society of the Pacific Conference Series, Vol. 228, Dynamics of Star Clusters and the Milky Way, ed. S. Deiters, B. Fuchs, A. Just, R. Spurzem, & R. Wielen, 61, doi: 10.48550/arXiv.astro-ph/0006438

    doi arXiv

  17. [26]

    C., Hurley, J

    Giersz, M., Heggie, D. C., Hurley, J. R., & Hypki, A. 2013, Monthly Notices of the Royal Astronomical Society, 431, 2184–2199, doi: 10.1093/mnras/stt307

    2013 doi

  18. [27]

    2015, MNRAS, 454, 3150, doi: 10.1093/mnras/stv2162

    Giersz, M., Leigh, N., Hypki, A., L¨ utzgendorf, N., & Askar, A. 2015, MNRAS, 454, 3150, doi: 10.1093/mnras/stv2162

    2015 doi

  19. [28]

    2003, Monthly Notices of the Royal Astronomical Society, 343, 781–795, doi: 10.1046/j.1365-8711.2003.06717.x

    Giersz, M., & Spurzem, R. 2003, Monthly Notices of the Royal Astronomical Society, 343, 781–795, doi: 10.1046/j.1365-8711.2003.06717.x

    2003

  20. [29]

    2015, in Journal of Physics Conference Series, Vol

    Gong, X., Lau, Y.-K., Xu, S., et al. 2015, in Journal of Physics Conference Series, Vol. 610, Journal of Physics Conference Series, 012011, doi: 10.1088/1742-6596/610/1/012011

    2015 doi

  21. [30]

    2024, arXiv e-prints, arXiv:2408.04603, doi: 10.48550/arXiv.2408.04603 19 H´ enon, M

    Hendriks, K., Zwick, L., & Samsing, J. 2024, arXiv e-prints, arXiv:2408.04603, doi: 10.48550/arXiv.2408.04603 19 H´ enon, M. 1971a, Ap&SS, 13, 284, doi: 10.1007/BF00649159 H´ enon, M. H. 1971b, Ap&SS, 14, 151, doi: 10.1007/BF00649201

    2024 doi arXiv

  22. [31]

    2018, ApJ, 856, 140, doi: 10.3847/1538-4357/aaafce

    Dosopoulou, F. 2018, ApJ, 856, 140, doi: 10.3847/1538-4357/aaafce

    2018 doi

  23. [32]

    R., Pols, O

    Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS, 315, 543, doi: 10.1046/j.1365-8711.2000.03426.x

    2000

  24. [33]

    R., Tout, C

    Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS, 329, 897, doi: 10.1046/j.1365-8711.2002.05038.x

    2002

  25. [34]

    2012, Monthly Notices of the Royal Astronomical Society, 429, 1221–1243, doi: 10.1093/mnras/sts415

    Hypki, A., & Giersz, M. 2012, Monthly Notices of the Royal Astronomical Society, 429, 1221–1243, doi: 10.1093/mnras/sts415

    2012 doi

  26. [35]

    2018, Progress of Theoretical and Experimental Physics, 2018, 073E01, doi: 10.1093/ptep/pty078

    Isoyama, S., Nakano, H., & Nakamura, T. 2018, Progress of Theoretical and Experimental Physics, 2018, 073E01, doi: 10.1093/ptep/pty078

    2018 doi

  27. [36]

    2012, Astronomy & Astrophysics, 538, A19, doi: 10.1051/0004-6361/201116923

    Jalali, B., Baumgardt, H., Kissler-Patig, M., et al. 2012, Astronomy & Astrophysics, 538, A19, doi: 10.1051/0004-6361/201116923

    2012 doi

  28. [37]

    C., Arumugasamy, P., Bagchi, M., et al

    Joshi, B. C., Arumugasamy, P., Bagchi, M., et al. 2018, Journal of Astrophysics and Astronomy, 39, 51, doi: 10.1007/s12036-018-9549-y

    2018 doi

  29. [38]

    2021, Progress of Theoretical and Experimental Physics, 2021, 05A105, doi: 10.1093/ptep/ptab019

    Kawamura, S., Ando, M., Seto, N., et al. 2021, Progress of Theoretical and Experimental Physics, 2021, 05A105, doi: 10.1093/ptep/ptab019

    2021 doi

  30. [39]

    Konstantinidis, S., Amaro-Seoane, P., & Kokkotas, K. D. 2013, A&A, 557, A135, doi: 10.1051/0004-6361/201219620

    2013 doi

  31. [40]

    2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x

    Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x

    2001

  32. [41]

    Lee, K. J. 2016, in Astronomical Society of the Pacific Conference Series, Vol. 502, Frontiers in Radio Astronomy and F AST Early Sciences Symposium 2015, ed. L. Qain & D. Li, 19

    2016

  33. [42]

    2024, arXiv e-prints, arXiv:2409.19665, doi: 10.48550/arXiv.2409.19665 LIGO Scientific Collaboration, & Virgo Collaboration

    Li, E.-K., Liu, S., Torres-Orjuela, A., et al. 2024, arXiv e-prints, arXiv:2409.19665, doi: 10.48550/arXiv.2409.19665 LIGO Scientific Collaboration, & Virgo Collaboration. 2019, Physical Review X, 9, 031040, doi: 10.1103/PhysRevX.9.031040 —. 2020, PhRvL, 125, 101102, doi: 10.1...

    2024 doi arXiv

  34. [43]

    J., & Brown, D

    Lindblom, L., Owen, B. J., & Brown, D. A. 2008, PhRvD, 78, 124020, doi: 10.1103/PhysRevD.78.124020

    2008 doi

  35. [44]

    2018, MNRAS, 473, 1719, doi: 10.1093/mnras/stx2531

    Lingam, M. 2018, MNRAS, 473, 1719, doi: 10.1093/mnras/stx2531

    2018 doi

  36. [45]

    2019, ApJ, 881, 41, doi: 10.3847/1538-4357/ab2dfb L¨ utzgendorf, N., Kissler-Patig, M., Noyola, E., et al

    Liu, B., Lai, D., & Wang, Y.-H. 2019, ApJ, 881, 41, doi: 10.3847/1538-4357/ab2dfb L¨ utzgendorf, N., Kissler-Patig, M., Noyola, E., et al. 2011, Astronomy & Astrophysics, 533, A36, doi: 10.1051/0004-6361/201116618

    2019 doi

  37. [46]

    N., & IPTA

    Manchester, R. N., & IPTA. 2013, Classical and Quantum Gravity, 30, 224010, doi: 10.1088/0264-9381/30/22/224010

    2013 doi

  38. [47]

    N., Hobbs, G., Bailes, M., et al

    Manchester, R. N., Hobbs, G., Bailes, M., et al. 2013, PASA, 30, e017, doi: 10.1017/pasa.2012.017

    2013 doi

  39. [48]

    A., Gair, J

    Mandel, I., Brown, D. A., Gair, J. R., & Miller, M. C. 2008, ApJ, 681, 1431, doi: 10.1086/588246 Mar ´ ın Pina, D., Gieles, M., Andrade, T., & Trani, A. A. 2025, arXiv e-prints, arXiv:2501.02907, doi: 10.48550/arXiv.2501.02907

    2008 doi arXiv

  40. [49]

    Martinez, M. A. S., Fragione, G., Kremer, K., et al. 2020, ApJ, 903, 67, doi: 10.3847/1538-4357/abba25

    2020 doi

  41. [50]

    2021, Progress of Theoretical and Experimental Physics, 2021, 05A107, doi: 10.1093/ptep/ptaa114

    Mei, J., Bai, Y.-Z., Bao, J., et al. 2021, Progress of Theoretical and Experimental Physics, 2021, 05A107, doi: 10.1093/ptep/ptaa114

    2021 doi

  42. [51]

    2001, ApJ, 556, 245, doi: 10.1086/321550 —

    Merritt, D. 2001, ApJ, 556, 245, doi: 10.1086/321550 —. 2002, ApJ, 568, 998, doi: 10.1086/339035

    2001 doi

  43. [52]

    2007, AJ, 133, 553, doi: 10.1086/510294

    Merritt, D., Berczik, P., & Laun, F. 2007, AJ, 133, 553, doi: 10.1086/510294

    2007 doi

  44. [53]

    2010, Astronomy and Astrophysics, 514, A52, doi: 10.1051/0004-6361/200912828

    Miocchi, P. 2010, Astronomy and Astrophysics, 514, A52, doi: 10.1051/0004-6361/200912828

    2010 doi

  45. [54]

    2008, ApJ, 676, 1008, doi: 10.1086/529002

    Noyola, E., Gebhardt, K., & Bergmann, M. 2008, ApJ, 676, 1008, doi: 10.1086/529002

    2008 doi

  46. [55]

    2010, ApJL, 719, L60, doi: 10.1088/2041-8205/719/1/L60

    Noyola, E., Gebhardt, K., Kissler-Patig, M., et al. 2010, ApJL, 719, L60, doi: 10.1088/2041-8205/719/1/L60

    2010 doi

  47. [56]

    2023, ApJ, 950, 3, doi: 10.3847/1538-4357/acce3b

    Peng, P., & Chen, X. 2023, ApJ, 950, 3, doi: 10.3847/1538-4357/acce3b

    2023 doi

  48. [57]

    Peters, P. C. 1964, Phys. Rev., 136, B1224, doi: 10.1103/PhysRev.136.B1224

    1964 doi

  49. [58]

    P., Naab, T., Spurzem, R., et al

    Rizzuto, F. P., Naab, T., Spurzem, R., et al. 2021, MNRAS, 501, 5257, doi: 10.1093/mnras/staa3634

    2021 doi

  50. [59]

    L., Morscher, M., Wang, L., et al

    Rodriguez, C. L., Morscher, M., Wang, L., et al. 2016, MNRAS, 463, 2109, doi: 10.1093/mnras/stw2121

    2016 doi

  51. [60]

    2021, A&A, 646, A20, doi: 10.1051/0004-6361/202039151

    Roupas, Z. 2021, A&A, 646, A20, doi: 10.1051/0004-6361/202039151

    2021 doi

  52. [61]

    S., & Schutz, B

    Sathyaprakash, B. S., & Schutz, B. F. 2009, Living Reviews in Relativity, 12, 2, doi: 10.12942/lrr-2009-2

    2009 doi

  53. [62]

    2022, PhRvD, 106, 064056, doi: 10.1103/PhysRevD.106.064056

    Sberna, L., Babak, S., Marsat, S., et al. 2022, PhRvD, 106, 064056, doi: 10.1103/PhysRevD.106.064056

    2022 doi

  54. [63]

    2023, Physical Review X, 13, 021035, doi: 10.1103/PhysRevX.13.021035

    Speri, L., Antonelli, A., Sberna, L., et al. 2023, Physical Review X, 13, 021035, doi: 10.1103/PhysRevX.13.021035

    2023 doi

  55. [64]

    Stodolkiewicz, J. S. 1986, AcA, 36, 19 20

    1986

  56. [65]

    J., Chatterjee, S., & Fragione, G

    Tiwari, A., Vijaykumar, A., Kapadia, S. J., Chatterjee, S., & Fragione, G. 2024a, arXiv e-prints, arXiv:2407.15117, doi: 10.48550/arXiv.2407.15117

    doi

  57. [66]

    2024b, MNRAS, 527, 8586, doi: 10.1093/mnras/stad3749

    Chatterjee, S. 2024b, MNRAS, 527, 8586, doi: 10.1093/mnras/stad3749

    doi

  58. [67]

    2024a, A VS Quantum Science, 6, 045002, doi: 10.1116/5.0219737 —

    Torres-Orjuela, A. 2024a, A VS Quantum Science, 6, 045002, doi: 10.1116/5.0219737 —. 2024b, arXiv e-prints, arXiv:2407.11293, doi: 10.48550/arXiv.2407.11293

    doi arXiv

  59. [68]

    2023, Phys

    Torres-Orjuela, A., & Chen, X. 2023, Phys. Rev. D, 107, 043027, doi: 10.1103/PhysRevD.107.043027

    2023 doi

  60. [69]

    2020, PhRvD, 101, 083028, doi: 10.1103/PhysRevD.101.083028

    Torres-Orjuela, A., Chen, X., & Amaro-Seoane, P. 2020, PhRvD, 101, 083028, doi: 10.1103/PhysRevD.101.083028

    2020 doi

  61. [70]

    2021, PhRvD, 104, 123025, doi: 10.1103/PhysRevD.104.123025

    Torres-Orjuela, A., Chen, X., & Amaro Seoane, P. 2021, PhRvD, 104, 123025, doi: 10.1103/PhysRevD.104.123025

    2021 doi

  62. [71]

    2024, Science China Physics, Mechanics, and Astronomy, 67, 259511, doi: 10.1007/s11433-023-2308-x

    Torres-Orjuela, A., Huang, S.-J., Liang, Z.-C., et al. 2024, Science China Physics, Mechanics, and Astronomy, 67, 259511, doi: 10.1007/s11433-023-2308-x

    2024 doi

  63. [72]

    2021, PhRvL, 126, 101105, doi: 10.1103/PhysRevLett.126.101105

    Toubiana, A., Sberna, L., Caputo, A., et al. 2021, PhRvL, 126, 101105, doi: 10.1103/PhysRevLett.126.101105

    2021 doi

  64. [73]

    A., Spera, M., Leigh, N

    Trani, A. A., Spera, M., Leigh, N. W. C., & Fujii, M. S. 2019, ApJ, 885, 135, doi: 10.3847/1538-4357/ab480a V´ azquez-Aceves, V., Amaro Seoane, P., Kuvatova, D., et al. 2023a, MNRAS, 518, 2113, doi: 10.1093/mnras/stac3286 V´ azquez-Aceves, V., Lin, Y., & Torres-Orjuela, A. 202...

    2019 doi arXiv

  65. [74]

    2015, MNRAS, 450, 4070, doi: 10.1093/mnras/stv817 —

    Wang, L., Spurzem, R., Aarseth, S., et al. 2015, MNRAS, 450, 4070, doi: 10.1093/mnras/stv817 —. 2016, MNRAS, 458, 1450, doi: 10.1093/mnras/stw274

    2015 doi

  66. [75]

    2023, PhRvL, 130, 241402, doi: 10.1103/PhysRevLett.130.241402

    Wardell, B., Pound, A., Warburton, N., et al. 2023, PhRvL, 130, 241402, doi: 10.1103/PhysRevLett.130.241402

    2023 doi

  67. [76]

    2003, ApJ, 598, 419, doi: 10.1086/378794

    Wen, L. 2003, ApJ, 598, 419, doi: 10.1086/378794

    2003 doi

  68. [77]

    2020, PhRvD, 102, 123009, doi: 10.1103/PhysRevD.102.123009

    Yu, H., Ma, S., Giesler, M., & Chen, Y. 2020, PhRvD, 102, 123009, doi: 10.1103/PhysRevD.102.123009

    2020 doi

  69. [78]

    2020, International Journal of Modern Physics D, 29, 1940005, doi: 10.1142/S0218271819400054

    Zhan, M.-S., Wang, J., Ni, W.-T., et al. 2020, International Journal of Modern Physics D, 29, 1940005, doi: 10.1142/S0218271819400054

    2020 doi

  70. [79]

    2020, MNRAS, 495, 2321, doi: 10.1093/mnras/staa1314

    Amaro-Seoane, P. 2020, MNRAS, 495, 2321, doi: 10.1093/mnras/staa1314

    2020 doi

  71. [80]

    R., Bortolas, E., et al

    Zwick, L., Capelo, P. R., Bortolas, E., et al. 2021, MNRAS, 506, 1007, doi: 10.1093/mnras/stab1818

    2021 doi

  72. [81]

    A., et al

    Zwick, L., Tiede, C., Trani, A. A., et al. 2024, PhRvD, 110, 103005, doi: 10.1103/PhysRevD.110.103005

    2024 doi

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