Pith. sign in

REVIEW 3 major objections 4 minor 60 references

Repeated lensed gravitational-wave images can fingerprint the spatial form of dark matter.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 05:20 UTC pith:CIQFGERG

load-bearing objection Solid, useful forecast; headline image thresholds are computed over all 311 images while only ~48% pass the stated wave-optics validity check at the reference FDM mass. the 3 major comments →

arxiv 2607.22787 v1 pith:CIQFGERG submitted 2026-07-24 astro-ph.CO gr-qc

Probing Dark Matter Substructure with Wave-Optics Distortions of Strongly Lensed LISA Gravitational Waves

Tonghua Liu , Kai Liao , Marek Biesiada , Jieci Wang This is my paper
classification astro-ph.CO gr-qc
keywords gravitational lensingwave opticsdark matter substructurefuzzy dark matterLISAgravitational wavesFermat potentiallensed images
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Strong lensing imprints the dark-matter structure of a galaxy on the phase of every gravitational-wave image, not just its arrival time and amplitude. This paper shows that after every lensed signal from a four-year LISA mission is refitted with the same smooth elliptical lens, the frequency-dependent part of the residual separates fuzzy dark matter (FDM) from ordinary Navarro–Frenk–White (NFW) and self-interacting (SIDM) halos. The separation reaches the 95% level with about 60 resolved image waveforms for NFW–FDM and about 110 for SIDM–FDM. If correct, a handful of repeated signals would let astronomers probe the spatial form of dark matter in lens galaxies, not just its total mass.

Core claim

The central claim is that the coherent density fluctuations of an FDM field leave a phase distortion that a smooth singular-isothermal-ellipsoid lens cannot absorb, while the slowly varying distortions of NFW and SIDM halos are largely degenerate with the constant, gradient, and curvature of the macro lens. Across the 132 lens systems and 311 time-resolved image waveforms of the simulated four-year LISA catalogue, the mismatch statistic separates the FDM population from NFW and SIDM at the 95% level when roughly 60 and 110 image waveforms are compared. The ordering is stable under repeated catalogue draws, and a single high-SNR image can recover the injected FDM coherence scale narrowly.

What carries the argument

The load-bearing device is the residual Fermat potential: around each macro image, the perturbing dark-matter potential is expanded and its constant term, gradient, and Hessian are subtracted, because those are the changes a smooth lens can already reproduce. What remains, δψ_res, enters the wave-optics amplification factor F_i(f) as a frequency-dependent phase and amplitude modulation. FDM's coherent density fluctuations survive this subtraction; NFW/SIDM's slowly varying potentials mostly do not.

Load-bearing premise

The load-bearing premise is that the local quadratic expansion used to compute the residual is valid for every image region included in the population statistic, even though only 48.2% of regions pass the stated validity check at the reference FDM mass.

What would settle it

Recompute the mismatch distribution and KS separation using only the image regions that pass the local-approximation validity criterion (f_pass = 48.2% at the reference mass). If the NFW–FDM and SIDM–FDM separation curves no longer reach 95% at 60 and 110 images, the central claim fails. Alternatively, a single high-SNR FDM image with known injected boson mass should yield a posterior concentrated at that mass; a wide or biased posterior would falsify the wave-optics model.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A four-year LISA mission with the assumed population would accumulate enough resolved images (60 to 110) to distinguish FDM from NFW/SIDM at 95% confidence.
  • The same separation holds when counts are done per lens system rather than per image: about 30 systems for NFW–FDM and 50 for SIDM–FDM.
  • The FDM signal is strongest for boson masses near 10⁻²¹ eV; at higher masses the short coherence length averages the projected fluctuation out over the image region.
  • A single high-SNR lensed image can measure the FDM coherence scale (boson mass) with a narrow posterior, conditional on fixed binary and lens parameters.
  • Varying the SIDM central-density factor from 1.5 to 5 changes the residual strength but keeps the SIDM population below the FDM level for the reference choice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A reader might test the robustness of the headline thresholds by recomputing the KS separation on the subset of image regions that pass the local-approximation validity condition (48.2% at the reference mass); if the curves move, the effective sample size is smaller than 311.
  • The same residual statistic could be applied to strongly lensed fast radio bursts or pulsar signals, where a frequency-dependent phase from dark-matter substructure would leave an analogous fingerprint, to check whether FDM-like coherence is unique to this wave band.
  • One could extend the population test by fitting all images from a single lens simultaneously, using the shared source and macro-lens parameters as additional constraints; this would likely reduce the required number of systems below the paper's per-image estimate.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper investigates whether strongly lensed gravitational-wave signals observed by LISA can distinguish between NFW, SIDM, and FDM dark-matter structures in the lens. The authors construct a simulated catalogue of 132 lens systems producing 311 time-resolved image waveforms, inject dark-matter perturbations, and then fit each waveform with a smooth SIE-plus-shear lens plus an eight-parameter binary waveform. The residual mismatch after this fit is used as a population statistic. They report that FDM leaves a larger frequency-dependent residual than NFW or SIDM, and that about 60 resolved image waveforms (NFW--FDM) or 110 (SIDM--FDM) are sufficient to distinguish the populations at the 95% level using a KS test. The paper includes extensive supplemental controls: a smooth-lens null range, SIDM profile variation, independent FDM density realizations, and a boson-mass scan.

Significance. If the quantitative claim survives scrutiny, this would be a genuinely new probe of dark-matter microphysics: repeated lensed LISA signals could constrain the spatial form of the lensing potential rather than only the total mass. The paper is careful in several respects: it runs a smooth-lens null control, checks the SIDM central-density dependence, tests three independent FDM density realizations, and distinguishes between image-level and system-level counting. The qualitative ordering NFW≈SIDM≪FDM is physically plausible and is supported by the controls. However, the headline thresholds are computed on a domain in which the stated wave-optics local approximation is unreliable for more than half of the FDM image regions at the reference mass. This is a load-bearing validity issue that must be addressed before the quantitative claim can be accepted.

major comments (3)
  1. [SM Sec. VIII; Fig. 3] The headline KS thresholds are computed on all 311 image waveforms, but at the reference mass mψ=10^-21 eV only f_pass=48.2% of image regions satisfy the local wave-optics validity condition used to derive F_i(f) from Eqs. (5)-(6). The paper itself states that the local approximation 'ceases to describe those regions reliably.' The all-sample FDM median in Fig. 3(a) is 1.12e-4, whereas the passing-sample median reported in Sec. VIII is 5.3e-4, confirming that invalid regions are included in the main statistic. Because the 60/110-image claim depends on the distribution of M over the full catalogue, the authors must either recompute Fig. 3 and the N95 thresholds on the passing subset, or provide a non-local estimate for the failing regions and demonstrate that their inclusion does not bias the KS separation. As written, the central calculation is unreliable for 51.8% of the images at the r
  2. [Main text Eqs. (5)-(6); SM Sec. IV] The amplitude and statistical properties of the FDM convergence fluctuation δκ_FDM are not specified quantitatively. The text says only that the amplitude is 'fixed by the host normalization,' with no equation, numerical value, or power spectrum. Since the entire FDM-vs-NFW/SIDM separation is driven by this fluctuation, an unspecified normalization is effectively a free parameter in the comparison; the statement that 'no additional FDM amplitude parameter was introduced' cannot be checked. Please provide the explicit normalization prescription and, ideally, a variation of the amplitude to show that the reported thresholds are not set by this choice.
  3. [SM Sec. VIII; main text Fig. 3] The validity fraction f_pass is defined only for FDM. The same local expansion of Eqs. (5)-(6) is used for NFW and SIDM, but no validity check is reported for those models. If a comparable fraction of NFW/SIDM regions are outside the domain of the local approximation, the KS comparison is not between equivalent calculations. Please report f_pass (or an equivalent diagnostic) for all three models, or justify why compact-subhalo perturbations always remain within the validity domain.
minor comments (4)
  1. [Fig. 3(b); Table I] The 95% thresholds are quoted as point values (60, 110, 30, 50) without uncertainty intervals. Since panel (b) shows variation under repeated catalogue draws, a bootstrap or percentile range on N95 would better represent the statistical uncertainty.
  2. [SM Sec. V] The prior-width parameter s=0.20 is described as having no physical interpretation, but the posterior and the mismatch at the posterior mode depend on it. A sentence on whether the quoted thresholds are stable under changes in s would be useful.
  3. [Eqs. (7) and (10)] The symbol M is used both for the perturber mass in Eq. (7) and for the mismatch M=1-R in Eq. (10). This notation conflict is confusing; consider using a different symbol for one of them.
  4. [SM Sec. VIII, Eq. (23)] The single-image boson-mass posterior is extremely narrow and is obtained with zero-noise injection and all nuisance parameters fixed. The text labels this as conditional, but the main text should state explicitly that this is not a realistic measurement; otherwise the quoted precision of 1.0006^{+0.0010}_{-0.0007}×10^-21 eV may be overinterpreted.

Circularity Check

0 steps flagged

No significant circularity: the waveform residual, population mismatch, and KS separation are computed quantities rather than definitions or self-citations.

full rationale

The claimed derivation chain is: model potentials -> local wave-optics integral F_i(f) (Eq. 6) -> smooth-lens waveform fit -> mismatch M (Eq. 10) -> KS separation (Fig. 3b). Each stage is an independent computation. The residual in Eq. (5) removes only the constant, gradient, and Hessian terms that a smooth lens can absorb; this is a projection defining the observable, not a definition of the conclusion that FDM leaves a larger residual. That conclusion is a numerical outcome of the NFW, SIDM, and FDM model potentials, not an identity. The FDM reference mass is chosen from the mass scan in Sec. VIII, which the paper explicitly states "Its role is to show why the reference mass was chosen and which part of the mass range supports the model-separation result"; this is a transparent forecast/parameter choice, not a fitted parameter renamed as a prediction. The single-event posterior in Eq. (23) is an injection-recovery calibration, not a prediction, and the paper states the width is conditional on fixed nuisance parameters. The main caveats — Sec. VIII reporting f_pass=48.2% at the reference mass and the local approximation ceasing "to describe those regions reliably" — are limitations of validity/robustness of the headline thresholds, not circular steps; they do not make the derivation equivalent to its inputs. No load-bearing self-citation or imported uniqueness theorem is present. Therefore, no circular step is identified.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The headline claim rests on standard lensing formalism plus several free input choices: the FDM boson mass (best-case value in the scan), the SIDM concentration factor, the subhalo fraction and mass function, and an unquantified FDM fluctuation amplitude. No invented entities are introduced; the models are pre-existing hypotheses. The load-bearing domain assumption is that subtracting constant/gradient/Hessian terms captures exactly what a smooth lens can absorb, and the paper's own f_pass = 48.2% shows the local expansion is marginal for about half the reference-mass sample.

free parameters (6)
  • FDM boson mass m_psi = 10^-21 eV (reference); 10^-22 to 10^-20 eV scanned
    Reference mass selected for the main comparison; Sec. VIII shows it yields the strongest residual, so the headline 60/110-image thresholds are best-case values conditional on this choice.
  • SIDM central-concentration factor B_c = 3 (reference); 1.5 and 5 in scan
    Hand-chosen 'moderately evolved' gravothermal branch (Eq. 14); not tied to a specific self-interaction cross section.
  • Projected substructure fraction f_sub = 5 x 10^-4
    Lens-plane bound-halo abundance; sets the overall NFW/SIDM population amplitude in the lens plane.
  • Subhalo mass function slope and range = dN/dM proportional to M^-1.9 over 10^5-10^7 M_sun
    Assumed CDM-subhalo population; anchors the projected substructure and the NFW/SIDM waveform perturbations.
  • FDM density-fluctuation amplitude = 'fixed by the host normalization' (value not stated)
    Sets the size of the FDM residual that the headline claim depends on; the normalization is never quantified in the text provided.
  • Bayesian prior-width parameter s = 0.20
    Sets prior ranges for the eight-parameter fit; the paper notes it has no physical interpretation, but it affects how much mismatch the smooth template can absorb.
axioms (5)
  • domain assumption Wave-optics diffraction integral (Eq. 6) gives the correct lensed waveform response to substructure.
    Standard Fresnel-Kirchhoff thin-lens formalism (Refs. [15-17, 26, 40]); assumed valid across the LISA band and the image-region integrations.
  • domain assumption Removing the constant a0, gradient a1, and Hessian A2 of the perturbing Fermat potential (Eq. 5) captures everything a smooth SIE-plus-shear lens can absorb.
    Load-bearing modeling choice: the residual that defines the FDM signal is whatever varies beyond this quadratic form. If a smooth lens can mimic higher-order terms, the FDM residual shrinks.
  • domain assumption The Ref. [39] population (132 lensed MBHB systems in four years) describes the real detectable LISA event rate.
    The 311-image sample and all population statistics inherit the source-population assumptions; line-of-sight halos and baryonic perturbers are excluded (acknowledged in Sec. IX).
  • domain assumption Zero-noise injection with the LISA noise-weighted likelihood reproduces real-data fitting behavior.
    Sec. V injects known signals d_q = h_q; the smooth-lens null range (Sec. II) is the control for this, but real noise realizations, selection effects, and calibration errors are absent.
  • domain assumption A two-sample KS test on repeated catalogue draws at p<0.05 measures mission-level distinguishability.
    The 95%-of-draws criterion (Fig. 3b) is a population statement, correctly labeled as such by the paper; it is an analysis choice rather than a physical law.

pith-pipeline@v1.3.0-alltime-deepseek · 15490 in / 22030 out tokens · 194653 ms · 2026-08-01T05:20:50.117938+00:00 · methodology

0 comments
read the original abstract

Strong lensing changes the phase of a gravitational-wave signal as well as its amplitude and arrival time. We study whether this phase information can distinguish three dark-matter structures in the lens: a Navarro--Frenk--White halo (NFW), a self-interacting halo (SIDM), and a fuzzy-dark-matter field (FDM). We generate waveforms for the detectable lensed massive-black-hole-binary population of a four-year LISA mission and fit every signal with the same smooth singular-isothermal-ellipsoid lens with external shear. In 132 lens systems, 311 images are resolved as separate signals in time. NFW and SIDM produce real waveform changes, but their slowly varying part is largely degenerate with the constant, gradient, and curvature of a smooth lens. The coherent density fluctuations of FDM leave a larger frequency-dependent residual after this fit. The NFW--FDM and SIDM--FDM populations become distinguishable with about 60 and 110 resolved image waveforms, respectively. These results show that repeated lensed LISA signals can probe the spatial form of dark matter in lens galaxies, rather than only the total lensing mass.

Figures

Figures reproduced from arXiv: 2607.22787 by Jieci Wang, Kai Liao, Marek Biesiada, Tonghua Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: gives a direct example of the effect mea￾sured below. The upper curves remain close because the dark-matter contribution is a perturbation to an already strongly magnified signal. Their differences are clearer in the lower panel, where the residual changes sign and oscillates with time. Such a residual is the time-domain form of a frequency-dependent change in Fi(f); it can￾not be described by changing onl… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the result. The blue points in panel (a) show every evaluated image region, while the green points retain the regions for which the local wave-optics approximation is valid. The distinction has a physical meaning. At low mψ, the coherence length is long and the adopted field normalization produces a large coher￾ent perturbation over many image regions. The local approximation then ceases to describe … view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 45 linked inside Pith

  1. [1]

    Bertone, D

    G. Bertone, D. Hooper, and J. Silk, Physics Reports405, 279 (2005), arXiv:hep-ph/0404175

    Pith/arXiv arXiv 2005

  2. [2]

    J. S. Bullock and M. Boylan-Kolchin, Annual Re- view of Astronomy and Astrophysics55, 343 (2017), arXiv:1707.04256 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  3. [3]

    D. J. E. Marsh, Physics Reports643, 1 (2016), arXiv:1510.07633 [astro-ph.CO]

    Pith/arXiv arXiv 2016

  4. [4]

    Tulin and H.-B

    S. Tulin and H.-B. Yu, Physics Reports730, 1 (2018), arXiv:1705.02358 [hep-ph]

    Pith/arXiv arXiv 2018

  5. [5]

    Mao and P

    S. Mao and P. Schneider, Monthly Notices of the Royal Astronomical Society295, 587 (1998)

    1998

  6. [6]

    R. B. Metcalf and P. Madau, The Astrophysical Journal 563, 9 (2001), astro-ph/0108224

    Pith/arXiv arXiv 2001

  7. [7]

    Dalal and C

    N. Dalal and C. S. Kochanek, The Astrophysical Journal 572, 25 (2002), astro-ph/0111456

    Pith/arXiv arXiv 2002

  8. [8]

    Vegetti and L

    S. Vegetti and L. V. E. Koopmans, Monthly Notices of the Royal Astronomical Society392, 945 (2009), arXiv:0805.0201 [astro-ph]

    Pith/arXiv arXiv 2009

  9. [9]

    Vegetti, L

    S. Vegetti, L. V. E. Koopmans, M. W. Auger, T. Treu, and A. S. Bolton, Monthly Notices of the Royal As- tronomical Society442, 2017 (2014), arXiv:1405.3666 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  10. [10]

    Vegetti, S

    S. Vegetti, S. Birrer, G. Despali, C. D. Fassnacht, D. Gilman, Y. Hezaveh, L. Perreault Levasseur, J. P. McKean, D. M. Powell, C. M. O’Riordan, and G. Vernardos, Space Science Reviews220, 58 (2024), arXiv:2306.11781 [astro-ph.CO]

    Pith/arXiv arXiv 2024

  11. [11]

    D.Gilman, S.Birrer, T.Treu, A.Nierenberg,andA.Ben- son, Monthly Notices of the Royal Astronomical Society 487, 5721 (2019), arXiv:1901.11031 [astro-ph.CO]

    Pith/arXiv arXiv 2019

  12. [12]

    Gilman, S

    D. Gilman, S. Birrer, A. Nierenberg, T. Treu, X. Du, and A. Benson, Monthly Notices of the Royal Astronom- ical Society491, 6077 (2020), arXiv:1908.06983 [astro- ph.CO]

    Pith/arXiv arXiv 2020

  13. [13]

    A. C. Şengül, A. Tsang, A. Diaz Rivero, C. Dvorkin, H.- M. Zhu, and U. Seljak, Physical Review D102, 063502 (2020), arXiv:2006.07383 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  14. [14]

    Gilman, S

    D. Gilman, S. Birrer, A. Nierenberg, and M. S. H. Oh, Monthly Notices of the Royal Astronomical Society533, 1687 (2024), arXiv:2403.03253 [astro-ph.CO]

    Pith/arXiv arXiv 2024

  15. [15]

    T. T. Nakamura and S. Deguchi, Progress of Theoretical Physics Supplement133, 137 (1999)

    1999

  16. [16]

    Takahashi and T

    R. Takahashi and T. Nakamura, The Astrophysical Jour- nal595, 1039 (2003), astro-ph/0305055

    Pith/arXiv arXiv 2003

  17. [17]

    Dai and T

    L. Dai and T. Venumadhav, arXiv e-prints , arXiv:1702.04724 (2017), arXiv:1702.04724 [gr-qc]

    Pith/arXiv arXiv 2017

  18. [18]

    M.Oguri,MonthlyNoticesoftheRoyalAstronomicalSo- ciety480, 3842 (2018), arXiv:1807.02584 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  19. [19]

    O. A. Hannuksela, K. Haris, K. K. Y. Ng, S. Kumar, A. K. Mehta, D. Keitel, T. G. F. Li, and P. Ajith, The Astrophysical Journal Letters874, L2 (2019), arXiv:1901.02674 [gr-qc]

    Pith/arXiv arXiv 2019

  20. [20]

    X. Shan, B. Hu, X. Chen, and R.-G. Cai, Nature Astron- omy9, 916 (2025)

    2025

  21. [21]

    Amaro-Seoaneet al., arXiv e-prints , arXiv:1702.00786 6 (2017), arXiv:1702.00786 [astro-ph.IM]

    P. Amaro-Seoaneet al., arXiv e-prints , arXiv:1702.00786 6 (2017), arXiv:1702.00786 [astro-ph.IM]

    Pith/arXiv arXiv 2017

  22. [22]

    Robson, N

    T. Robson, N. J. Cornish, and C. Liu, Classical and Quantum Gravity36, 105011 (2019), arXiv:1803.01944 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  23. [23]

    Kleinet al., Physical Review D93, 024003 (2016), arXiv:1511.05581 [gr-qc]

    A. Kleinet al., Physical Review D93, 024003 (2016), arXiv:1511.05581 [gr-qc]

    Pith/arXiv arXiv 2016

  24. [24]

    Tamanini, C

    N. Tamanini, C. Caprini, E. Barausse, A. Sesana, A. Klein, and A. Petiteau, Journal of Cosmology and Astroparticle Physics2016(4), 002, arXiv:1601.07112 [astro-ph.CO]

    Pith/arXiv arXiv

  25. [25]

    Marsat, J

    S. Marsat, J. G. Baker, and T. Dal Canton, Physical Review D103, 083011 (2021), arXiv:2003.00357 [gr-qc]

    Pith/arXiv arXiv 2021

  26. [26]

    Çalışkan, N

    M. Çalışkan, N. A. Kumar, L. Ji, J. M. Ezquiaga, R. Cotesta, E. Berti, and M. Kamionkowski, Physical Review D108, 123543 (2023), arXiv:2307.06990 [astro- ph.CO]

    Pith/arXiv arXiv 2023

  27. [27]

    Tambalo, M

    G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.- Y. Cheung, Physical Review D108, 103529 (2023), arXiv:2212.11960 [astro-ph.CO]

    Pith/arXiv arXiv 2023

  28. [28]

    Ando, arXiv e-prints , arXiv:2603.04267 (2026), arXiv:2603.04267 [astro-ph.CO]

    S. Ando, arXiv e-prints , arXiv:2603.04267 (2026), arXiv:2603.04267 [astro-ph.CO]

    arXiv 2026

  29. [29]

    Ando, arXiv e-prints , arXiv:2606.21519 (2026), arXiv:2606.21519 [astro-ph.CO]

    S. Ando, arXiv e-prints , arXiv:2606.21519 (2026), arXiv:2606.21519 [astro-ph.CO]

    Pith/arXiv arXiv 2026

  30. [30]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, The Astrophysical Journal490, 493 (1997), arXiv:astro- ph/9611107

    arXiv 1997

  31. [31]

    Diemer and A

    B. Diemer and A. V. Kravtsov, The Astrophysical Jour- nal799, 108 (2015), arXiv:1407.4730 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  32. [32]

    D. N. Spergel and P. J. Steinhardt, Physical Review Let- ters84, 3760 (2000), arXiv:astro-ph/9909386

    Pith/arXiv arXiv 2000

  33. [33]

    Schive, T

    H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature Physics10, 496 (2014), arXiv:1406.6586 [astro-ph.GA]

    Pith/arXiv arXiv 2014

  34. [34]

    J. H. H. Chan, H.-Y. Schive, T.-P. Woo, and T. Chiueh, Monthly Notices of the Royal Astronomical Society478, 2686 (2018), arXiv:1712.01947 [astro-ph.GA]

    Pith/arXiv arXiv 2018

  35. [35]

    Laroche, D

    A. Laroche, D. Gilman, X. Li, J. Bovy, and X. Duan, Monthly Notices of the Royal Astronomical Society517, 1867 (2022), arXiv:2206.11269 [astro-ph.CO]

    Pith/arXiv arXiv 2022

  36. [36]

    L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Physi- cal Review D95, 043541 (2017), arXiv:1610.08297 [astro- ph.CO]

    Pith/arXiv arXiv 2017

  37. [37]

    S.Singh, G.BrandodeOliveira, S.Savastano,andM.Zu- malacárregui, Journal of Cosmology and Astroparticle Physics2025(07), 025, arXiv:2502.10758 [astro-ph.CO]

    Pith/arXiv arXiv

  38. [38]

    J. Liu, K. Liao, and Y. Gong, Physical Review D110, 083536 (2024), arXiv:2405.04779 [astro-ph.CO]

    Pith/arXiv arXiv 2024

  39. [39]

    Gutiérrez and M

    J. Gutiérrez and M. Lagos, Physical Review D112, 123512 (2025), arXiv:2510.02061 [astro-ph.CO]

    arXiv 2025

  40. [40]

    Villarrubia-Rojo, S

    H. Villarrubia-Rojo, S. Savastano, M. Zumalacárregui, L. Choi, S. Goyal, L. Dai, and G. Tambalo, Physical Re- view D111, 103539 (2025), arXiv:2409.04606 [gr-qc]

    Pith/arXiv arXiv 2025

  41. [41]

    Schneider, J

    P. Schneider, J. Ehlers, and E. E. Falco,Gravitational Lenses(Springer-Verlag, Berlin, 1992)

    1992

  42. [42]

    Kormann, P

    R. Kormann, P. Schneider, and M. Bartelmann, Astron- omy and Astrophysics284, 285 (1994)

    1994

  43. [43]

    C. R. Keeton, C. S. Kochanek, and U. Seljak, The Astrophysical Journal482, 604 (1997), arXiv:astro- ph/9610163

    arXiv 1997

  44. [44]

    Birrer and A

    S. Birrer and A. Amara, Physics of the Dark Universe 22, 189 (2018), arXiv:1803.09746 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  45. [45]

    Birreret al., Journal of Open Source Software6, 3283 (2021), arXiv:2106.05976 [astro-ph.CO]

    S. Birreret al., Journal of Open Source Software6, 3283 (2021), arXiv:2106.05976 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  46. [46]

    London, S

    L. London, S. Khan, E. Fauchon-Jones, C. García, M.Hannam, S.Husa, X.Jiménez-Forteza, C.Kalaghatgi, F. Ohme, and F. Pannarale, Physical Review Letters 120, 161102 (2018), arXiv:1708.00404 [gr-qc]

    Pith/arXiv arXiv 2018

  47. [47]

    M. L. Katz, S. Marsat, A. J. K. Chua, S. Babak, and S. L. Larson, Physical Review D102, 023033 (2020), arXiv:2005.01827 [gr-qc]

    Pith/arXiv arXiv 2020

  48. [48]

    Cutler, Physical Review D57, 7089 (1998), gr- qc/9703068

    C. Cutler, Physical Review D57, 7089 (1998), gr- qc/9703068

    arXiv 1998

  49. [49]

    N. J. Cornish and L. J. Rubbo, Physical Review D67, 022001 (2003), gr-qc/0209011

    Pith/arXiv arXiv 2003

  50. [50]

    Ashton, M

    G. Ashton, M. Hübner, P. D. Lasky, C. Talbot, K. Ack- ley, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov, F. Hernandez Vivanco, J. Harms, M. E. Lower, G. D. Meadors, D. Melchor, E. Payne, M. Pitkin, J. Powell, N. Sarin, R. J. E. Smith, and E. Thrane, The Astrophysical Journal Supplement Series241, 27 (2019), arXiv:1811.02042 [astro-ph.IM]

    Pith/arXiv arXiv 2019

  51. [51]

    Storn and K

    R. Storn and K. Price, Journal of Global Optimization 11, 341 (1997)

    1997

  52. [52]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, Publications of the Astronomical Society of the Pa- cific125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    Pith/arXiv arXiv 2013

  53. [53]

    Bohéet al., Physical Review D95, 044028 (2017), arXiv:1611.03703 [gr-qc]

    A. Bohéet al., Physical Review D95, 044028 (2017), arXiv:1611.03703 [gr-qc]

    Pith/arXiv arXiv 2017

  54. [54]

    LIGO Scientific Collaboration, LALSuite: LIGO Scien- tific Collaboration Algorithm Library (2018)

    2018

  55. [55]

    S. Yang, X. Du, Z. C. Zeng, A. Benson, F. Jiang, E. O. Nadler, and A. H. G. Peter, Journal of Cosmology and Astroparticle Physics2022(9), 054, arXiv:2205.02957 [astro-ph.CO]

    Pith/arXiv arXiv

  56. [56]

    E. O. Nadler, D. Yang, and H.-B. Yu, The Astrophys- ical Journal Letters958, L39 (2023), arXiv:2306.01830 [astro-ph.CO]

    Pith/arXiv arXiv 2023

  57. [57]

    S. Hou, D. Yang, N. Li, and G. Li, arXiv e-prints , arXiv:2502.14964 (2025), arXiv:2502.14964 [astro- ph.CO]. Supplemental Material I. Population Construction The source layer follows the detectable lensed LISA massive-black-hole-binary population in Ref. [39], which gives 132 lensed systems over the mission duration. The macro-lens layer is regenerated w...

    Pith/arXiv arXiv 2025

  58. [58]

    the image signal-to-noise ratio must satisfy the de- tection criterion used in the source population

  59. [59]

    the system must have a detected partner with 1 week<∆t <1 yr

  60. [60]

    The final sample has 132 lens systems and 311 resolved image waveforms

    accepted images in the same system must be sepa- rated by more thanmax(7 days, Tsig). The final sample has 132 lens systems and 311 resolved image waveforms. The distribution of usable images per system is 103 doubles, 11 triples, and 18 quadruples. The 7 distinction between systems and image waveforms is im- portant: the former is the astrophysical event...