pith. sign in

arxiv: 2606.08899 · v1 · pith:Q57LRSWZnew · submitted 2026-06-08 · 🌌 astro-ph.CO · astro-ph.GA· astro-ph.HE

Mock Catalogs of Strongly Lensed Gravitational Waves via a Halo Model Approach with Space-borne Detectors

Mingqi Sun , Kai Liao , Youkai Li , Tonghua Liu , Hengyu Wu , Shaoqi Hou , Tao Yang , Xilong Fan
show 1 more author
Marek Biesiada
This is my paper

Pith reviewed 2026-06-27 15:56 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GAastro-ph.HE
keywords strongly lensed gravitational wavesLISADECIGOmock catalogshalo modelmassive black hole binariessignal overlapspace-borne detectors
0
0 comments X

The pith

Halo model mocks predict LISA could detect 0 to 131 strongly lensed gravitational waves in four years.

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

The paper builds a simulation framework that combines halo models for intervening lenses with astrophysical models for source populations to produce mock catalogs of strongly lensed gravitational-wave events for space-borne detectors. It reports that four years of LISA data should contain between zero and 131 lensed massive black hole binary events, corresponding to a lensing probability as high as 0.3 percent, while one year of DECIGO data yields zero to 44 lensed stellar-mass binary events at roughly 0.15 percent probability. The work further establishes that multiple images of the same event frequently overlap in time, altering measured signal-to-noise ratios and complicating event identification. A reader would care because these catalogs supply the first quantitative forecasts needed to design analysis methods before the instruments collect data.

Core claim

Based on realistic astrophysical models for both the source population and the lens distribution, we construct mock catalogs of lensed GW events. For a four-year LISA observation, the expected number of lensed events ranges from 0 to 131, depending on the adopted formation model of massive black hole binaries. The corresponding lensing probability for MBHBs can reach up to ∼0.3%. For DECIGO, the number of lensed events in a one-year observation is expected to lie in the range of 0--44, with a lensing probability of ∼0.15% for stellar-mass binary black holes, binary neutron stars, and neutron star--black hole binaries. We further show that the overlap of lensed signals is a common feature in

What carries the argument

Halo model approach to the lens distribution integrated with source population models to generate the GW-LMC-Space mock catalogs of strongly lensed events.

If this is right

  • Four years of LISA data are expected to contain between zero and 131 lensed massive black hole binary events.
  • The lensing probability for massive black hole binaries can reach approximately 0.3 percent.
  • One year of DECIGO data should contain between zero and 44 lensed stellar-mass binary events at roughly 0.15 percent probability.
  • Overlapping lensed signals occur commonly enough to change signal-to-noise ratio estimates and event identification procedures.

Where Pith is reading between the lines

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

  • The wide range in predicted event numbers implies that actual detections will help discriminate among competing massive black hole binary formation models.
  • Analysis pipelines will need to incorporate methods that handle simultaneous arrival of multiple images rather than treating each image in isolation.
  • The mock catalogs can be used to test and calibrate algorithms for identifying strong lensing in real gravitational-wave data streams.
  • Longer observation campaigns or joint analyses across detectors would likely increase the total yield of identifiable lensed events beyond the quoted single-detector figures.

Load-bearing premise

The chosen astrophysical models for source populations and the halo model for lens distributions accurately represent reality and can be used directly to generate the mock catalogs.

What would settle it

A four-year LISA data set containing a number of strongly lensed events well outside the 0-131 range, or a DECIGO data set outside the 0-44 range, or an absence of overlapping images in any confirmed multiply-imaged events.

Figures

Figures reproduced from arXiv: 2606.08899 by Hengyu Wu, Kai Liao, Marek Biesiada, Mingqi Sun, Shaoqi Hou, Tao Yang, Tonghua Liu, Xilong Fan, Youkai Li.

Figure 1
Figure 1. Figure 1: SNR distribution (encoded in color map) of the intrinsic population of MBHBs for different formation models in 4-year LISA observation. The ”x” axis is the total mass of the system, while the ”y” axis denotes the redshift of the source (merging MBHB). For each model, we randomly selected 1000 events, considered their event rate from the intrinsic population and ploted the SNR distribution. If flow > fISCO,… view at source ↗
Figure 2
Figure 2. Figure 2: The six formation models all exhibit a broad distribution of signal durations, indicating that the rela￾tionship between signal duration and lensing time delay must be carefully treated when computing the SNR of lensed events. We therefore adopt the following strategy: 1. If the lensing time delay is shorter than the GW signal duration, the lensed signals overlap in time and are treated as a single combine… view at source ↗
Figure 3
Figure 3. Figure 3: Time delay distribution of the lensed events for different formation models in 100,000 samples of LISA obser￾vation. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping. obtain more robust statistical information of the lensed event rates. To account for the stochasticity of th… view at source ↗
Figure 4
Figure 4. Figure 4: We present the distribution of lens redshift zl (redshift of the host halo) and source redshift zs for 105 samples, together with a scatter plot of the maximum SNR ratio rmax (defined as the ratio of the SNR of the overlapped signal to that of the unlensed signal) for different formation models in a four-year LISA observation. The color bar indicates the value of rmax, while the blue and green dashed lines… view at source ↗
Figure 5
Figure 5. Figure 5: Time delay difference distribution combined with SNR ratio distribution in the 100,000 samples in LISA obser￾vation. Different colors represent different formation models. where F+(θS, ϕS, ψS) = 1 2 (1 + cos2 θS) cos 2ϕS cos 2ψS − cos θS sin 2ϕS sin 2ψS , F×(θS, ϕS, ψS) = 1 2 (1 + cos2 θS) cos 2ϕS sin 2ψS + cos θS sin 2ϕS cos 2ψS . (18) are antenna patterns. The (¯θS, ϕ¯ S) denote the source position in th… view at source ↗
Figure 6
Figure 6. Figure 6: The joint distribution of the time delay difference and the magnification ratio for the model HSnodSN in 100,000 samples of LISA observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The antenna pattern function of DECIGO in 1yr observation, with the source position fixed at ¯θS = ϕ¯S = π/4 and the orientation of the orbital angular momentum fixed at ˆθL = ϕˆL = 0. of the DCO considered. H(z) is the expansion rate in ΛCDM model considered in this paper. The source pop￾ulation of DCOs studied in this paper, following our previous work (Li et al. 2026) is the same as consid￾ered in (Pi´o… view at source ↗
Figure 8
Figure 8. Figure 8: SNR distributions of the intrinsic populations of BBHs, BNSs, and NSBHs for the DECIGO case over a one-year observation period. In each subplot, the SNR distribution is divided into three regimes to distinguish different SNR ranges [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the gravitational-wave signal du￾ration for BBHs, BNSs, and NSBHs during a one-year DE￾CIGO mission. The durations are calculated for all events in each source type. corporating the potential overlap of lensed signals based on their durations and lensing time delays. The number of lensed events without imposing an SNR threshold for each source type is presented in Ta￾ble 8. The lensed SNR i… view at source ↗
Figure 10
Figure 10. Figure 10: Time delay distribution of the lensed events for BBHs, BNSs, and NSBHs in the DECIGO band. The upper panel shows the time delay distribution without considering the overlapping of lensed signals, while the lower panel shows the distribution when we consider the overlapping. For the detected lensed events, we summarize the sta￾tistical results for each source type when signal overlap is taken into account … view at source ↗
Figure 11
Figure 11. Figure 11: The distribution of lens redshift zl and the source redshift zs for all samples in 1-yr observation with the scatter plot of the max SNR ratio rmax for BBHs, BNSs, and NSBHs in the DECIGO band. The color bar represents the value of rmax while the blue and green dashed lines represent the median of the source redshift and lens redshift, respectively [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time delay difference distribution combined with SNR ratio distribution in 1-year observation of DE￾CIGO. Different colors represent different model types. predicts that most events have SNR < 8, indicating a substantial discrepancy between models. A similar level of variation is found in the predicted event rates. The HSnodnoSN model yields 39,364 intrinsic events over a four-year LISA observation, where… view at source ↗
Figure 13
Figure 13. Figure 13: The joint distribution of the time delay differ￾ence and the magnification ratio for the source BBH in 1-year DECIGO observation. The triple and quintuple plots (blue and orange points) represent the distribution of the baseline 3-image and 5-image systems, which are already shown in the [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

Future space-borne gravitational-wave (GW) detectors, such as LISA and DECIGO, are expected to detect a large number of GW events, a fraction of which may be strongly lensed by intervening galaxies or galaxy clusters. In this work, we develop a comprehensive framework to simulate strongly lensed GWs in the context of space-borne detectors. Based on realistic astrophysical models for both the source population and the lens distribution, we construct mock catalogs of lensed GW events, referred to as \textbf{GW-LMC-Space}. Our results show that, for a four-year LISA observation, the expected number of lensed events ranges from $0$ to $131$, depending on the adopted formation model of massive black hole binaries (MBHBs). The corresponding lensing probability for MBHBs can reach up to $\sim 0.3\%$. For DECIGO, we find that the number of lensed events in a one-year observation is expected to lie in the range of $0$--$44$, with a lensing probability of $\sim 0.15\%$ for stellar-mass binary black holes (BBHs), binary neutron stars (BNSs), and neutron star--black hole binaries (NSBHs). We further show that the overlap of lensed signals is a common feature in space-borne detectors, which can significantly affect both the signal-to-noise ratio (SNR) estimation and event identification. These results highlight the importance of accounting for signal overlap in the analysis of strongly lensed GW events in future space-borne GW observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a halo-model framework for lens distributions combined with astrophysical source-population models to generate mock catalogs (GW-LMC-Space) of strongly lensed GW events detectable by LISA and DECIGO. It reports that, for a 4-year LISA mission, the expected number of lensed MBHB events ranges from 0 to 131 depending on the MBHB formation model, with lensing probability up to ~0.3%; for a 1-year DECIGO mission the corresponding numbers are 0–44 with ~0.15% probability for stellar-mass binaries. The work additionally shows that overlapping lensed signals are common and affect SNR estimation and event identification.

Significance. If the input models are representative, the generated mock catalogs constitute a concrete resource for developing and testing lensing-search pipelines for space-borne detectors. The explicit demonstration that signal overlap is a generic feature of long-duration space-borne waveforms supplies a practical motivation for SNR and identification algorithms that incorporate this effect. The transparent conditioning of all numerical results on the choice of formation models is a methodological strength.

minor comments (2)
  1. [Abstract] Abstract: the quoted ranges (0–131, 0–44) are not accompanied by the number or identities of the formation models that produce the extremes; adding this information would make the dependence on model choice immediately quantifiable.
  2. The manuscript states that overlap affects SNR estimation but does not quantify the typical fractional change in SNR or the fraction of events for which overlap occurs; a short table or figure summarizing these statistics would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on mock catalogs of strongly lensed GW events for space-borne detectors and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs mock catalogs by feeding external astrophysical models (cited formation scenarios for MBHBs, BBHs, BNSs, NSBHs and a halo-model lens distribution) into a simulation pipeline; the reported ranges (0-131 events for LISA, 0-44 for DECIGO) and probabilities are direct numerical outputs of that pipeline conditioned on the choice of input models. No equation or result is shown to be equivalent to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The overlap statement is a qualitative consequence of signal duration rather than a derived quantity. The derivation is therefore self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, preventing a complete audit of parameters and assumptions. The quoted ranges rest on the choice of MBHB formation models and the halo model for lenses.

free parameters (1)
  • MBHB formation model choice
    Different models produce the full 0-131 range, indicating these are external inputs that control the output counts.
axioms (1)
  • domain assumption Halo model provides a sufficiently accurate description of the intervening galaxy/cluster distribution for strong-lensing optical depth calculations of GWs.
    The framework is explicitly based on a halo model approach for the lens distribution.

pith-pipeline@v0.9.1-grok · 5854 in / 1241 out tokens · 22779 ms · 2026-06-27T15:56:21.706170+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Finite-Core Signatures in LISA-Band Wave-Optics Lensing by Low-Mass Dark Matter Halos

    astro-ph.CO 2026-06 unverdicted novelty 5.0

    Finite cores in low-mass dark matter halos produce distinct complex residuals in LISA-band wave-optics amplification that cannot be fully mimicked by lower-concentration NFW profiles and peak at rc/rs ≃ 0.25-0.3.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · cited by 1 Pith paper · 7 internal anchors

  1. [1]

    The Science of the Einstein Telescope

    Abac, A., Abramo, R., Albanesi, S., et al. 2025, arXiv e-prints, arXiv:2503.12263, doi: 10.48550/arXiv.2503.12263

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2503.12263 2025

  2. [2]

    P., Abbott, R., Abbott, T

    Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, PhRvL, 116, 061102, doi: 10.1103/PhysRevLett.116.061102

    work page doi:10.1103/physrevlett.116.061102 2016

  3. [3]

    T., Oguri, M., Birrer, S., et al

    Abe, K. T., Oguri, M., Birrer, S., et al. 2025, The Open Journal of Astrophysics, 8, 8, doi: 10.33232/001c.128482

    work page doi:10.33232/001c.128482 2025

  4. [4]

    Laser Interferometer Space Antenna

    Amaro-Seoane, P., Audley, H., Babak, S., et al. 2017, arXiv e-prints, arXiv:1702.00786, doi: 10.48550/arXiv.1702.00786

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1702.00786 2017

  5. [5]

    2015, ApJ, 812, 72, doi: 10.1088/0004-637X/812/1/72

    Antonini, F., Barausse, E., & Silk, J. 2015, ApJ, 812, 72, doi: 10.1088/0004-637X/812/1/72

    work page doi:10.1088/0004-637x/812/1/72 2015

  6. [6]

    , keywords =

    Barausse, E. 2012, MNRAS, 423, 2533, doi: 10.1111/j.1365-2966.2012.21057.x

    work page doi:10.1111/j.1365-2966.2012.21057.x 2012

  7. [7]

    2023, PhRvD, 108, 103034, doi: 10.1103/PhysRevD.108.103034

    Barausse, E., Dey, K., Crisostomi, M., et al. 2023, PhRvD, 108, 103034, doi: 10.1103/PhysRevD.108.103034

    work page doi:10.1103/physrevd.108.103034 2023

  8. [8]

    2020a, ApJ, 904, 16, doi: 10.3847/1538-4357/abba7f

    Bonetti, M. 2020a, ApJ, 904, 16, doi: 10.3847/1538-4357/abba7f

    work page doi:10.3847/1538-4357/abba7f

  9. [9]

    2020b, General Relativity and Gravitation, 52, 81, doi: 10.1007/s10714-020-02691-1

    Barausse, E., Berti, E., Hertog, T., et al. 2020b, General Relativity and Gravitation, 52, 81, doi: 10.1007/s10714-020-02691-1

    work page doi:10.1007/s10714-020-02691-1

  10. [10]

    H., Hearin , A

    Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182

    work page doi:10.1093/mnras/stz1182 2019

  11. [11]

    S., Wechsler, R

    Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013, ApJL, 762, L31, doi: 10.1088/2041-8205/762/2/L31

    work page doi:10.1088/2041-8205/762/2/l31 2013

  12. [12]

    2023, MNRAS, 518, 5298, doi: 10.1093/mnras/stac3052

    Biscoveanu, S., Landry, P., & Vitale, S. 2023, MNRAS, 518, 5298, doi: 10.1093/mnras/stac3052

    work page doi:10.1093/mnras/stac3052 2023

  13. [13]

    K.-W., et al

    Cousins, B., Schumacher, K., Chung, A. K.-W., et al. 2026, PhRvL, 136, 101003, doi: 10.1103/4lzh-bm7y

    work page doi:10.1103/4lzh-bm7y 2026

  14. [14]

    2018, ApJS, 239, 35, doi: 10.3847/1538-4365/aaee8c

    Diemer, B. 2018, ApJS, 239, 35, doi: 10.3847/1538-4365/aaee8c

    work page doi:10.3847/1538-4365/aaee8c 2018

  15. [15]

    2019, ApJ, 871, 168, doi: 10.3847/1538-4357/aafad6

    Diemer, B., & Joyce, M. 2019, ApJ, 871, 168, doi: 10.3847/1538-4357/aafad6

    work page doi:10.3847/1538-4357/aafad6 2019

  16. [16]

    Diemer, B., & Kravtsov, A. V. 2015, ApJ, 799, 108, doi: 10.1088/0004-637X/799/1/108

    work page doi:10.1088/0004-637x/799/1/108 2015

  17. [17]

    2013, ApJ, 779, 72, doi: 10.1088/0004-637X/779/1/72

    Dominik, M., Belczynski, K., Fryer, C., et al. 2013, ApJ, 779, 72, doi: 10.1088/0004-637X/779/1/72

    work page doi:10.1088/0004-637x/779/1/72 2013

  18. [18]

    A Horizon Study for Cosmic Explorer: Science, Observatories, and Community

    Evans, M., Adhikari, R. X., Afle, C., et al. 2021, arXiv e-prints, arXiv:2109.09882, doi: 10.48550/arXiv.2109.09882 Guti´ errez, J., & Lagos, M. 2025, PhRvD, 112, 123512, doi: 10.1103/yd5h-ql5f

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2109.09882 2021

  19. [19]

    2019, PhRvD, 100, 084002, doi: 10.1103/PhysRevD.100.084002

    Hohmann, M., J¨ arv, L., Krˇ sˇ s´ ak, M., & Pfeifer, C. 2019, PhRvD, 100, 084002, doi: 10.1103/PhysRevD.100.084002

    work page doi:10.1103/physrevd.100.084002 2019

  20. [20]

    2021, Galaxies, 9, 14, doi: 10.3390/galaxies9010014

    Ishikawa, T., Iwaguchi, S., Michimura, Y., et al. 2021, Galaxies, 9, 14, doi: 10.3390/galaxies9010014

    work page doi:10.3390/galaxies9010014 2021

  21. [21]

    2020, MNRAS, 492, 3662, doi: 10.1093/mnras/staa069

    Ishiyama, T., & Ando, S. 2020, MNRAS, 492, 3662, doi: 10.1093/mnras/staa069

    work page doi:10.1093/mnras/staa069 2020

  22. [22]

    2016, PhRvD, 93, 024003, doi: 10.1103/PhysRevD.93.024003 LensedGW

    Klein, A., Barausse, E., Sesana, A., et al. 2016, PhRvD, 93, 024003, doi: 10.1103/PhysRevD.93.024003 LensedGW. 2026, LensedGW/GW-LMC-Space: GW-LMC-Space, v2.0, Zenodo, doi: 10.5281/zenodo.20540878

    work page doi:10.1103/physrevd.93.024003 2016

  23. [23]

    2025, PhRvD, 112, 023546, doi: 10.1103/5kk7-7f4n

    Lester, E., & Bolejko, K. 2025, PhRvD, 112, 023546, doi: 10.1103/5kk7-7f4n

    work page doi:10.1103/5kk7-7f4n 2025

  24. [24]

    Mock Catalogs of Strongly Lensed Gravitational Waves via A Halo Model Approach with Ground-based Detectors

    Li, Y., Liao, K., Sun, M., et al. 2026, arXiv e-prints, arXiv:2603.09289, doi: 10.48550/arXiv.2603.09289 18Sun et al

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.09289 2026

  25. [25]

    2017, Nature Communications, 8, 1148, doi: 10.1038/s41467-017-01152-9

    Liao, K., Fan, X.-L., Ding, X., Biesiada, M., & Zhu, Z.-H. 2017, Nature Communications, 8, 1148, doi: 10.1038/s41467-017-01152-9

    work page doi:10.1038/s41467-017-01152-9 2017

  26. [26]

    2016, Classical and Quantum Gravity, 33, 035010, doi: 10.1088/0264-9381/33/3/035010

    Luo, J., Chen, L.-S., Duan, H.-Z., et al. 2016, Classical and Quantum Gravity, 33, 035010, doi: 10.1088/0264-9381/33/3/035010 Macci` o, A. V., Dutton, A. A., van den Bosch, F. C., et al. 2007, Mon. Not. R. Astron. Soc., 378, 55, doi: 10.1111/j.1365-2966.2007.11720.x

    work page doi:10.1088/0264-9381/33/3/035010 2016

  27. [27]

    G., & Canton, T

    Marsat, S., Baker, J. G., & Canton, T. D. 2021, PhRvD, 103, 083011, doi: 10.1103/PhysRevD.103.083011

    work page doi:10.1103/physrevd.103.083011 2021

  28. [28]

    D., & Silk, J

    Mukherjee, S., Wandelt, B. D., & Silk, J. 2020, MNRAS, 494, 1956, doi: 10.1093/mnras/staa827

    work page doi:10.1093/mnras/staa827 2020

  29. [29]

    F., Frenk, C

    Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563, doi: 10.1086/177173

    work page doi:10.1086/177173 1996

  30. [30]

    2018, MNRAS, 480, 3842, doi: 10.1093/mnras/sty2145 —

    Oguri, M. 2018, MNRAS, 480, 3842, doi: 10.1093/mnras/sty2145 —. 2021, PASP, 133, 074504, doi: 10.1088/1538-3873/ac12db

    work page doi:10.1093/mnras/sty2145 2018

  31. [31]

    2020, ApJ, 901, 58, doi: 10.3847/1538-4357/abafab

    Oguri, M., & Takahashi, R. 2020, ApJ, 901, 58, doi: 10.3847/1538-4357/abafab

    work page doi:10.3847/1538-4357/abafab 2020

  32. [32]

    2020, MNRAS, 496, 2591, doi: 10.1093/mnras/staa1479

    Okabe, T., Oguri, M., Peirani, S., et al. 2020, MNRAS, 496, 2591, doi: 10.1093/mnras/staa1479

    work page doi:10.1093/mnras/staa1479 2020

  33. [33]

    C., & Mathews, J

    Peters, P. C., & Mathews, J. 1963, Physical Review, 131, 435, doi: 10.1103/PhysRev.131.435 Pi´ orkowska-Kurpas, A., Hou, S., Biesiada, M., et al. 2021, ApJ, 908, 196, doi: 10.3847/1538-4357/abd482 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910

    work page doi:10.1103/physrev.131.435 1963

  34. [34]

    2020, International Journal of Modern Physics A, 35, 2050075, doi: 10.1142/S0217751X2050075X

    Ruan, W.-H., Guo, Z.-K., Cai, R.-G., & Zhang, Y.-Z. 2020, International Journal of Modern Physics A, 35, 2050075, doi: 10.1142/S0217751X2050075X

    work page doi:10.1142/s0217751x2050075x 2020

  35. [35]

    2010, PhRvL, 105, 251101, doi: 10.1103/PhysRevLett.105.251101

    Sereno, M., Sesana, A., Bleuler, A., et al. 2010, PhRvL, 105, 251101, doi: 10.1103/PhysRevLett.105.251101

    work page doi:10.1103/physrevlett.105.251101 2010

  36. [36]

    2025, PhRvD, 112, 023002, doi: 10.1103/xll3-68j2

    Sun, M., & Liao, K. 2025, PhRvD, 112, 023002, doi: 10.1103/xll3-68j2

    work page doi:10.1103/xll3-68j2 2025

  37. [37]

    Detection of Multiband Lensed Gravitational Waves from Dark Matter Halos with Deep Learning

    Sun, M., Wu, J., Li, J., et al. 2025, arXiv e-prints, arXiv:2511.09107, doi: 10.48550/arXiv.2511.09107

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2511.09107 2025

  38. [38]

    2003, ApJ, 595, 1039, doi: 10.1086/377430 The LIGO Scientific Collaboration, The Virgo

    Takahashi, R., & Nakamura, T. 2003, ApJ, 595, 1039, doi: 10.1086/377430 The LIGO Scientific Collaboration, The Virgo

    work page doi:10.1086/377430 2003

  39. [39]

    GWTC-4.0: Population Properties of Merging Compact Binaries

    Collaboration, The KAGRA Collaboration, Abac, A. G., et al. 2025a, GWTC-4.0: Population Properties of Merging Compact Binaries. https://arxiv.org/abs/2508.18083 The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration, et al. 2025b, arXiv e-prints, arXiv:2508.18082, doi: 10.48550/arXiv.2508.18082 —. 2025c, arXiv e-prints, arXiv:...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2508.18082

  40. [40]

    V., Klypin, A., et al

    Tinker, J., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709, doi: 10.1086/591439 van der Wel, A., Martorano, M., H¨ außler, B., et al. 2024, ApJ, 960, 53, doi: 10.3847/1538-4357/ad02ee

    work page internal anchor Pith review doi:10.1086/591439 2008

  41. [41]

    2011, PhRvD, 83, 044011, doi: 10.1103/PhysRevD.83.044011

    Yagi, K., & Seto, N. 2011, PhRvD, 83, 044011, doi: 10.1103/PhysRevD.83.044011

    work page doi:10.1103/physrevd.83.044011 2011

  42. [42]

    2020, MNRAS, 497, L56, doi: 10.1093/mnrasl/slaa107

    Yang, T., Birrer, S., & Hu, B. 2020, MNRAS, 497, L56, doi: 10.1093/mnrasl/slaa107

    work page doi:10.1093/mnrasl/slaa107 2020

  43. [43]

    2019, ApJ, 880, 50, doi: 10.3847/1538-4357/ab271e

    Yang, T., Hu, B., Cai, R.-G., & Wang, B. 2019, ApJ, 880, 50, doi: 10.3847/1538-4357/ab271e

    work page doi:10.3847/1538-4357/ab271e 2019

  44. [44]

    2025, arXiv e-prints, arXiv:2505.09507, doi: 10.48550/arXiv.2505.09507

    Ying, X., & Yang, T. 2025, arXiv e-prints, arXiv:2505.09507, doi: 10.48550/arXiv.2505.09507

    work page doi:10.48550/arxiv.2505.09507 2025