Probability of gravitational-wave lensing by intermediate-mass black holes and globular clusters

Jose Mar\'ia Ezquiaga; Lorenz Zwick; Luka Vujeva; Rico K. L. Lo

REVIEW 3 major objections 4 minor 298 references

Probability of gravitational-wave lensing by intermediate-mass black holes and globular clusters

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that strong gravitational-wave lensing by intermediate-mass black holes in globular clusters is rare—at most 1 in 1000 strong-lens events—and that the candidate event GW231123 is therefore unlikely to be such a lens (odds…

desk verdict Careful, honest rate calculation for GC-IMBH lensing of strongly lensed GWs; the numbers hold up, the caveats are stated, and only minor polish is needed. read the letter →

arxiv 2608.07667 v1 pith:6WUSNW6E submitted 2026-08-07 astro-ph.CO gr-qc

Luka Vujeva , Jose María Ezquiaga , Rico K. L. Lo , Lorenz Zwick This is my paper · ORCID

classification astro-ph.COgr-qc

keywords gravitationalwaveslensingintermediate-massblackholesglobularclustersopticaldepthdarkmattersubstructureprimordialGW231123

topics Dark Matter

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 paper computes how often a gravitational wave that is already strongly lensed by a galaxy is further lensed by an intermediate-mass black hole (IMBH) embedded in a globular cluster. The central result is that this compound-lensing rate is at most 1 in 1000 strong-lens events, falling to 1 in 10,000 under the fiducial optimistic assumptions, and approaching only $\sim 1\%$ for very high magnifications ($\mu>100$) that are themselves rare. The calculation is driven by the optical depth of globular clusters inside the strong-lens image annuli, using a lens model that stacks a singular isothermal sphere halo, a singular isothermal sphere cluster, and a point-mass IMBH with external convergence and shear. Because the rate is so low, the paper concludes that the candidate event GW231123, whose inferred lens mass is about $1000\,M_\odot$, is very unlikely to be an IMBH+GC lens: the odds ratio against it is $2.80\times10^{-12}$. The low rate also means lensed gravitational waves remain a clean probe of dark-matter substructure and primordial black holes, whose lensing rates can be orders of magnitude higher.

What carries the argument

The load-bearing object is the compound lens model of Eq. (A1), which combines a truncated singular isothermal sphere (SIS) halo, an SIS globular cluster, and a point-mass IMBH, all subject to external convergence and shear. The lens equation for this potential is solved to find image positions, magnifications, and arrival-time differences, and the lensing criterion demands two images with time delay $\Delta t<1$ s and relative magnification $\mu_{\rm rel}>0.1$. The cross section $\hat{\sigma}_{\rm GC}$ for this sub-lensing is integrated against the Sérsic surface-density profile of the globular-cluster system over the annulus in the image plane that the main halo's magnification threshold selects, producing the optical-depth ratio $\tau_{\rm GC}/\tau_{\rm halo}$ that defines the conditional probability of Eq. (25).

What would settle it

Take the next hundred strongly lensed gravitational-wave events and count how many contain a sub-second image pair with a lens mass near $1000\,M_\odot$: the fiducial model predicts fewer than 0.1, so a single clean detection would put the rate above the predicted $\le 10^{-3}$ and falsify the GC-hosted IMBH assumption.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the conditional probability $P^{\rm Lens}_{\rm GC}(\tau_{\rm GC}|\tau_{\rm halo})$ — the fraction of galaxy-scale strongly lensed gravitational-wave events that would also be distorted by an IMBH in a globular cluster — is suppressed to the range $10^{-4}$ to $10^{-2}$ depending on the magnification threshold, with the fiducial sub-second time-delay criterion yielding $\lesssim 10^{-3}$. The suppression is geometric and robust: most of the optical depth comes from clusters near the main halo's critical curves, where the external convergence and shear reshape the caustics, and the sub-second delay requirement cuts the rate by about an order of magnitude relative to a tens-of-minutes threshold. Applying this rate as an astrophysical prior to GW231123 gives a Bayes factor of $1/166$ and an odds ratio of $2.80\times10^{-12}$ against the IMBH+GC lensing hypothesis, even though GW231123 is the strongest lensing candidate seen so far. The authors therefore state that lensed GWs are unlikely to be confused with known astrophysical potentials and can serve as clean probes of sub-galactic dark-matter substructure and PBHs.

Load-bearing premise

The calculation assumes that intermediate-mass black holes live at the centers of globular clusters, with cluster abundances and sizes following observed scaling relations, and that other IMBH populations (wandering halo IMBHs, dwarf-galaxy nuclei) are negligible; if many such black holes float free in galaxy halos, the lensing rate could be higher than reported.

Editorial extensions

If this is right

The fraction of galaxy-scale strong-lens events that also carry an IMBH+GC sub-lens is $\lesssim 10^{-3}$, so searches for repeated chirps with sub-second delays can treat this known-astrophysics background as negligible.
GW231123, the strongest lensed-GW candidate to date, is disfavored as an IMBH+GC lens by an odds ratio of $2.80\times10^{-12}$, reinforcing the conclusion that its lensing interpretation needs another lens population or a non-lensing explanation.
In next-generation ground-based detectors, the projected number of detectable IMBH+GC compound-lens events remains below 1, so any such detection would be a major surprise and point to a richer IMBH population than modeled.
Uniformly distributed primordial black holes with a dark-matter fraction above $\sim 10^{-6}$ would lens more GW events than IMBHs in globular clusters, making strongly lensed GWs a sensitive probe of PBH abundances.

Reading between the lines

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

If the unconstrained wandering-IMBH population (10–100 per halo in some simulations) is real, the compound-lensing rate could exceed the quoted upper bound, so the paper's numbers are best read as a floor on the clean-probe argument rather than a hard ceiling on all IMBH lensing.
A quick empirical test: re-analyze strongly lensed events allowing time delays up to tens of minutes; the model predicts the IMBH+GC rate would rise by about an order of magnitude toward $\sim 10^{-3}$, so a null result at that threshold tightens the bound while a detection demands a different lens class.
Cluster-scale strong lenses, which produce higher magnifications and more images, are the most promising sites to hunt for IMBH/GC sub-lensing because the conditional probability peaks near the main halo's critical curves, even though their absolute event rate is small.
The odds-ratio estimate ignores selection-effect differences between the lensed and non-lensed hypotheses, so the headline $2.80\times10^{-12}$ should be treated as an astrophysical-prior-informed statement whose exact value could shift under a full hierarchical selection model.

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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

Read it. The headline: this is a clean, careful rate calculation for a specific compound lensing channel—GWs already strongly lensed by a galaxy being further lensed by a GC-hosted IMBH—and the first to do it in a population sense and apply it to GW231123. The optical depth machinery is standard but the application is new, and the validation against lenstronomy plus the robustness tests (Sersic index, virial radius, time-delay threshold) give me confidence the numbers are what they say. The result—conditional rates at the 10^-3 to 10^-4 level, dropping to 10^-4 or below in the fiducial optimistic case—is credible and makes the point that this channel is unlikely to contaminate dark-matter substructure searches. The odds ratio of 2.8e-12 against GW231123 being lensed by a GC-IMBH is a useful quantitative anchor.

The soft spots are minor. The abstract's 'at most 1/1000' vs 'approaches 1% at mu>100' is a wording problem, not a numbers problem: 1% is the rate within a high-magnification subsample; the overall rate stays below 10^-3. They should spell that out in the abstract. Eq. C7 in the appendix has a duplicated p(z_L|z_S,H_L) factor—clearly a typo, but it should be fixed. The rate estimates have no formal error bars; the robustness tests bracket the systematics at factors of 0.7–3, which is honest but should be stated as the error budget.

On the stress-test note: the exclusion of wandering IMBHs is a real caveat, and the paper says so. They explicitly frame the calculation as conditional on the GC-hosted population being the relevant one, and they give reasons (dwarf nuclei have small cross sections; wandering IMBHs are unconstrained). The 'at most 1/1000' is an upper limit within that model, not a universal bound on all IMBH lensing. If a wandering population at 10-100 per halo is substantial, the rates shift, and the GW231123 odds ratio would move too. That doesn't make the calculation wrong, but it means the 'disfavoring' conclusion is model-dependent. If they want to claim a general bound, they'd need to bracket the wandering population quantitatively; right now they just wave at it. Worth asking for in revision.

This paper deserves a serious referee. It's a useful, honest, reproducible piece of rate estimation that sharpens the interpretation of lensed GW candidates. I'd cite it and would take it to the reading group.

Referee Report

3 major / 4 minor

Summary. The paper estimates the rate at which gravitational waves that are already strongly lensed by galaxy-scale halos are additionally lensed by intermediate-mass black holes (IMBHs) hosted in globular clusters (GCs). The authors build a forward model with a Sersic distribution of GCs, truncated SIS models for both halo and GC, a central point-mass IMBH, and external convergence and shear, and they validate the composite lens solver against lenstronomy. They compute optical depths and a conditional lensing probability as functions of the external magnification threshold and the time-delay threshold, test robustness to Sersic index, virial radius, and other assumptions, and forecast detectable event numbers for current and future detectors. Applying the resulting astrophysical prior to the candidate GW231123, they report a Bayes factor of 1/166 and an odds ratio of 2.80e-12 against the IMBH+GC lensing interpretation, concluding that lensed GWs should be a clean probe of dark matter substructure and primordial black holes.

Significance. If the calculation holds, it supplies a much-needed astrophysical prior for the interpretation of lensed gravitational-wave candidates and for forecasts of substructure lensing. The paper's strengths are its explicit forward-modeling setup, the validation against lenstronomy, and the robustness tests on Sersic index, virial radius, and time-delay thresholds. The main conclusions, however, are stated more strongly than the calculation supports: the headline bound is threshold-dependent, and the rate estimate covers only GC-hosted IMBHs. These are fixable framing issues rather than fundamental flaws, and the odds-ratio application to GW231123 is useful and internally consistent once the scope is made explicit.

major comments (3)

[Abstract and Sec. 4, Fig. 2] The abstract's statement that the relative rate is 'at most 1/1000' contradicts the threshold-dependent result shown in Fig. 2 and stated in Sec. 4: P_Lens_GC varies from roughly 10^-4 at an external magnification threshold of 2 to roughly 10^-2 at a threshold of 100. Thus for mu > 100 the rate is about 1%, not bounded by 1/1000. Please report the threshold-dependent values or explicitly state that 1/1000 is the bound after averaging over the magnification distribution of detectable strongly lensed events; as written, the headline number and the figure disagree.
[Sec. 2 and Sec. 5 / Appendix C] The conclusion that GW231123 is unlikely to be lensed by an IMBH is based on the GC-hosted population only. Section 2 acknowledges that wandering IMBHs from tidal stripping and minor mergers are observationally unconstrained and may number 10-100 per halo, but no quantitative upper bound on their lensing optical depth is given. Because the prior odds in Eq. (C9) is constructed from the GC-hosted rate alone, the reported odds ratio of 2.80e-12 and the statement 'disfavoring such an interpretation for GW231123' are conditional on the wandering population being subdominant. Please add an order-of-magnitude estimate for this population or explicitly qualify the abstract and conclusions as applying only to GC-hosted IMBHs.
[Sec. 3, Eq. (24) with Sec. 2, Eq. (1)] The occupation fraction f_occ is not visibly present in the central optical-depth integral. Eq. (24) integrates the GC surface density Sigma_GC(R), which is normalized to N_GC, but does not multiply by f_occ; if f_occ=1 is assumed throughout (as the Fig. 3 caption suggests), state this explicitly, and if f_occ<1 is varied, either include the factor in Eq. (24) or define sigma_GC as including it. Without this, the calculation cannot be reproduced and the meaning of 'fiducial optimistic scenario' is ambiguous.

minor comments (4)

[Appendix C, Eq. (C7)] The factor p(zL|zS,H_L) is duplicated in the integrand; one factor should be removed or replaced by the intended source-redshift prior p(zS). Please correct the displayed equation.
[Sec. 2] The statement that dwarf-galaxy nuclei are 'negligible' as IMBH hosts is asserted without a quantitative comparison; a one-sentence estimate of their optical depth relative to the GC-hosted population would strengthen the justification.
[Sec. 5] The claim that uniformly distributed PBHs overtake the IMBH+GC rate for a dark matter fraction larger than roughly 1e-6 is not derived or referenced in the text; please add the calculation or a citation.
[Appendix C, Eq. (C9)] The notation P_HL/HNL for the prior odds is easily confused with the odds ratio O_HL/HNL; consider renaming the prior odds (for example, pi_HL/HNL) to avoid ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the rate estimate is a forward model from external empirical GC scaling relations, and the GW231123 odds ratio uses the model as a prior rather than as a fit.

full rationale

The central quantity, P_Lens_GC = tau_GC/tau_halo (Eq. 25), is computed by integrating the observed GC surface-density profile (Eqs. 6-10) together with a composite SIS plus point-mass lens potential (Appendix A) that is validated against lenstronomy and analytic limits (An & Evans 2006; Finch et al. 2002). The input parameters f_occ and f_mass are stated astrophysical assumptions, not fitted to the target rate or to GW231123, and the paper explicitly explores their variation in the robustness tests of Sec. 4. The odds ratio in Appendix C uses the model's predicted rate as a prior odds (Eq. C9) and the GW231123 parameter-estimation samples as data, which is standard Bayesian model comparison rather than a prediction that reduces to its input. The lensing criteria Delta t < 1 s and mu_rel > 0.1 are justified by an external parameter-estimation study (Chan et al. 2026b), and the posterior samples come from an independent analysis (Chan et al. 2026a); although several authors overlap with the present paper, these citations provide data products and detection criteria, not the rate law itself. The exclusion of wandering IMBHs is an explicit modeling assumption with a stated caveat and does not constitute a circular step. No equation in the paper is equivalent by construction to the claimed result, and the headline rate is not obtained by renaming or refitting any input quantity. The skeptical concern about unmodeled IMBH populations affects model coverage and astrophysical robustness, not circularity.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The central rate depends on a set of adopted astrophysical parameters (occupation fraction, mass ratio, GC profile) and detection criteria. None of these are fitted to the target conclusion; they are either from observations or varied as robustness tests. The main assumptions are clearly stated, but several (f_occ, f_mass, GC SIS model) are unconstrained astrophysically.

free parameters (10)

f_occ = 1 (fiducial optimistic)
Fraction of GCs hosting an IMBH; set to unity as an optimistic ceiling, though the true occupation fraction is unconstrained.
f_mass = 0.1 (fiducial)
IMBH mass as a fraction of GC mass; used in Fig. 2; rates are insensitive to this ratio.
GC Sersic index n = 1-3 (varied)
Index of the GC surface density profile; varied in robustness tests, changing rates by factors 0.7-3.
GC half-light radius = 3 pc
Adopted from Milky Way GC population; used to fix the SIS virial radius at 6 pc.
Time delay threshold = 1 s
Definition of overlapping images; changing it to tens of minutes raises the rate to ~10^-3.
Relative magnification threshold = 0.1
Minimum ratio for the second image to be associated with the same event, from Chan et al. 2026b.
Source redshift z_S = 2 (fiducial, 0.5-3 varied)
Fiducial source redshift; rates are weakly dependent on z_S.
External magnification threshold mu0 = 2-100 (varied)
Minimum magnification of the galaxy-scale lens; drives the rate from 10^-4 to 10^-2.
Halo mass range = 10^11-10^14 M_sun
Integration range for the Schechter halo mass function.
Detection SNR threshold = 8
SNR per image for a lensed detection in the forecasting.

assumptions (7)

domain assumption GCs are modeled as truncated singular isothermal spheres (SIS) with virial radius twice the half-mass radius.
Used in Sec. 2 to relate GC mass, size, and velocity dispersion; simplified from King/Plummer profiles.
domain assumption The IMBH is a point-mass lens exactly at the GC center.
Sec. 3; offset IMBHs are not considered.
domain assumption The main halo is a truncated SIS with kappa_ext = gamma_ext, and external convergence/shear are derived from the halo.
Sec. 3, Eqs. 11-16.
standard math The compound lens potential is the sum of the SIS, point mass, external shear, and external convergence.
Appendix A, Eq. A1; standard gravitational lensing.
domain assumption A detectable second image requires time delay < 1 s and relative magnification > 0.1.
Sec. 3, based on Chan et al. 2026b and the need for overlapping signals.
domain assumption The GC population scaling relations (Eqs. 3, 4, 8) hold at the lens redshifts and halo masses considered, with no redshift evolution.
Sec. 2; redshift evolution is omitted as subdominant.
domain assumption Geometric optics applies to the IMBH/GC lensing, so magnifications and time delays are computed from the lens equation.
Throughout; wave-optics distortions are discussed qualitatively but not included in the rate calculation.

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Cite this review

Pith. "Pith review of Probability of gravitational-wave lensing by intermediate-mass black holes and globular clusters." pith.science (2026). https://pith.science/paper/6WUSNW6E

@misc{pith2026260807667,
  author       = {Pith},
  title        = {Pith review of: Probability of gravitational-wave lensing by intermediate-mass black holes and globular clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WUSNW6E}},
  note         = {Machine review of arXiv:2608.07667}
}

abstract

Strongly lensed gravitational waves (GWs) are powerful probes of substructure in the lens. Intermediate-mass black holes (IMBHs) are postulated to be efficient lenses that may distort the lensed waveforms of currently detectable stellar-mass compact binaries, as hinted by GW231123. Assuming that IMBHs are located in globular clusters (GCs), we compute the rate at which they would affect strongly lensed repeated chirps produced by galaxy-scale lenses, considering a compound lens system. Exploring different astrophysical model assumptions and lensing criteria, we find that the relative rate is at most 1/1000 and decays to 1/10,000 for our fiducial optimistic scenario. At high magnifications, $\mu$ > 100, the relative rate approaches 1%, but these cases are intrinsically rare in absolute value. Our results imply that GW lensing by IMBHs and GCs is unlikely, disfavoring such an interpretation for GW231123. In turn, they point towards lensed GWs being a clean probe of dark matter substructures and primordial black holes.

Figures

Figures reproduced from arXiv: 2608.07667 by the authors.

**Figure 1.** Schematic view of the problem. A GW is strongly lensed by a galaxy producing repeated chirps (colored circles in the image plane at arcsecond separations, left panel). We study the probability that one of the repeated copies is further distorted by the lensing produced by an IMBH embedded in a GC and subject to the external convergence of the main halo (inset plots at hundreds of micro-arcseconds). The cyan shaded r… view at source ↗

**Figure 2.** (Left) Conditional probability of lensing for M•/MGC = 0.1 shown against the magnification threshold. The solid line corresponds to the probability for GCs outside of the main halo critical curves, and the dashed lines correspond to those inside the critical curves. Shaded regions show the extent of the probability over the source redshift range zS ∈ [0.5, 3.0]. The individual optical depths of the GCs and the main … view at source ↗

**Figure 3.** Projection for the number of lensed events as a function of time for our fiducial model with an IMBH occupation fraction in GCs of 1. The number of events lensed by the compound IMBH and GC configurations are found to always lie well below 1 detection. For reference, we also show the number of expected GWs and strongly lensed GWs. Projections include uncertainties in the observed BBH merger rate. the host galaxy).… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Inconsistency of the inferred lensing parameters µrel, ∆t from GW231123 with the prediction from our lensing by IMBH+GC setup. The probability density p(µrel, ∆t|zL, zS) from our simulations, marginalized over the mass ratio M•/MGC, is computed at some fiducial lens a…

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