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Evolution of precessing binary black holes on eccentric orbits using orbit-averaged evolution equations

Khun Sang Phukon1,*, Nathan K. Johnson-McDaniel2,†, Amitesh Singh2,‡, and Anuradha Gupta2,§

  • *Contact author: k.s.phukon@bham.ac.uk
  • †Contact author: nkjm.physics@gmail.com
  • ‡Contact author: amiteshsingh487@gmail.com
  • §Contact author: agupta1@olemiss.edu
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Phys. Rev. D 112, 044067 – Published 29 August, 2025

DOI: https://doi.org/10.1103/2yxp-pcbw

Abstract

The most general bound binary black hole system has an eccentric orbit and precessing spins. The detection of such a system with significant eccentricity close to the merger would be a clear signature of dynamical formation. In order to study such systems, it is important to be able to evolve their spins and eccentricity from the larger separations at which the binary formed to the smaller separations at which it is detected, or vice versa. Knowledge of the precessional evolution of the binary’s orbital angular momentum can also be used to twist up aligned-spin eccentric waveform models to create a spin-precessing eccentric waveform model. In this paper, we present a new publicly available code to evolve eccentric, precessing binary black holes using orbit-averaged post-Newtonian (PN) equations from the literature. The spin-precession dynamics is 2PN accurate, i.e., with the leading spin-orbit and spin-spin corrections. The evolution of orbital parameters (orbital frequency, eccentricity, and periastron precession), which follow the quasi-Keplerian parametrization, is 3PN accurate in the point particle terms and includes the leading order spin-orbit and spin-spin effects. All the spin-spin terms include the quadrupole-monopole interaction. The eccentricity enhancement functions in the fluxes use the high-accuracy hyperasymptotic expansions from Loutrel and Yunes [Classical Quantum Gravity 34, 044003 (2017)]. We discuss various features of the code and study the evolution of the orbital and spin-precession parameters of eccentric, precessing binary black holes. In particular, we study the dependence of the spin morphologies on eccentricity, where we find that the transition point from one spin morphology to another can depend nonmonotonically on eccentricity, and the fraction of binaries in a given morphology at a given point in the evolution of a population depends on the instantaneous eccentricity.

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Corrections

14 January, 2026

Correction: The DOI given in Ref. [126] has been changed.

Article Text

References (129)

  1. J. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32, 074001 (2015).
  2. F. Acernese et al. (Virgo Collaboration), Classical Quantum Gravity 32, 024001 (2015).
  3. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. X 13, 041039 (2023).
  4. A. H. Nitz, S. Kumar, Y.-F. Wang, S. Kastha, S. Wu, M. Schäfer, R. Dhurkunde, and C. D. Capano, Astrophys. J. 946, 59 (2023).
  5. S. Olsen, T. Venumadhav, J. Mushkin, J. Roulet, B. Zackay, and M. Zaldarriaga, Phys. Rev. D 106, 043009 (2022).
  6. A. K. Mehta, S. Olsen, D. Wadekar, J. Roulet, T. Venumadhav, J. Mushkin, B. Zackay, and M. Zaldarriaga, Phys. Rev. D 111, 024049 (2025).
  7. D. Wadekar, J. Roulet, T. Venumadhav, A. K. Mehta, B. Zackay, J. Mushkin, S. Olsen, and M. Zaldarriaga, arXiv:2312.06631.
  8. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Astrophys. J. Lett. 970, L34 (2024).
  9. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. X 13, 011048 (2023).
  10. I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Astrophys. J. Lett. 921, L31 (2021).
  11. I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Astrophys. J. 940, 171 (2022).
  12. I. M. Romero-Shaw, P. D. Lasky, E. Thrane, and J. C. Bustillo, Astrophys. J. Lett. 903, L5 (2020).
  13. V. Gayathri, J. Healy, J. Lange, B. O’Brien, M. Szczepanczyk, I. Bartos, M. Campanelli, S. Klimenko, C. O. Lousto, and R. O’Shaughnessy, Nat. Astron. 6, 344 (2022).
  14. N. Gupte et al., arXiv:2404.14286.
  15. M. d. L. Planas, A. Ramos-Buades, C. García-Quirós, H. Estellés, S. Husa, and M. Haney, arXiv:2504.15833.
  16. M. Hannam et al., Nature (London) 610, 652 (2022).
  17. E. Payne, S. Hourihane, J. Golomb, R. Udall, D. Davis, and K. Chatziioannou, Phys. Rev. D 106, 104017 (2022).
  18. R. Macas, A. Lundgren, and G. Ashton, Phys. Rev. D 109, 062006 (2024).
  19. G. Morras, G. Pratten, and P. Schmidt, arXiv:2503.15393.
  20. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Astrophys. J. Lett. 915, L5 (2021).
  21. G. Morras, G. Pratten, and P. Schmidt, Phys. Rev. D 111, 084052 (2025).
  22. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 125, 101102 (2020).
  23. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Astrophys. J. Lett. 900, L13 (2020).
  24. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. D 109, 022001 (2024).
  25. X. Liu, Z. Cao, and Z.-H. Zhu, Classical Quantum Gravity 41, 195019 (2024).
  26. R. Gamba, D. Chiaramello, and S. Neogi, Phys. Rev. D 110, 024031 (2024).
  27. X. Liu, Z. Cao, and L. Shao, Phys. Rev. D 101, 044049 (2020).
  28. A. Ramos-Buades, A. Buonanno, M. Khalil, and S. Ossokine, Phys. Rev. D 105, 044035 (2022).
  29. V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Phys. Rev. Res. 1, 033015 (2019).
  30. J. Calderón Bustillo, N. Sanchis-Gual, A. Torres-Forné, and J. A. Font, Phys. Rev. Lett. 126, 201101 (2021).
  31. I. M. Romero-Shaw, D. Gerosa, and N. Loutrel, Mon. Not. R. Astron. Soc. 519, 5352 (2023).
  32. P. C. Peters, Phys. Rev. 136, B1224 (1964).
  33. A. Tucker and C. M. Will, Phys. Rev. D 104, 104023 (2021).
  34. I. Kowalska, T. Bulik, K. Belczynski, M. Dominik, and D. Gondek-Rosinska, Astron. Astrophys. 527, A70 (2011).
  35. L. Wen, Astrophys. J. 598, 419 (2003).
  36. R. M. O’Leary, B. Kocsis, and A. Loeb, Mon. Not. R. Astron. Soc. 395, 2127 (2009).
  37. Y.-B. Bae, C. Kim, and H. M. Lee, Mon. Not. R. Astron. Soc. 440, 2714 (2014).
  38. F. Antonini, S. Chatterjee, C. L. Rodriguez, M. Morscher, B. Pattabiraman, V. Kalogera, and F. A. Rasio, Astrophys. J. 816, 65 (2016).
  39. K. Silsbee and S. Tremaine, Astrophys. J. 836, 39 (2017).
  40. J. Samsing, Phys. Rev. D 97, 103014 (2018).
  41. C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F. A. Rasio, Phys. Rev. Lett. 120, 151101 (2018).
  42. M. Zevin, I. M. Romero-Shaw, K. Kremer, E. Thrane, and P. D. Lasky, Astrophys. J. Lett. 921, L43 (2021).
  43. M. Dall’Amico, M. Mapelli, S. Torniamenti, and M. A. Sedda, Astron. Astrophys. 683, A186 (2024).
  44. J. Fuller and L. Ma, Astrophys. J. Lett. 881, L1 (2019).
  45. K. Belczynski et al., Astron. Astrophys. 636, A104 (2020).
  46. D. Gerosa and E. Berti, Phys. Rev. D 95, 124046 (2017).
  47. M. Fishbach, D. E. Holz, and B. Farr, Astrophys. J. Lett. 840, L24 (2017).
  48. V. Baibhav and V. Kalogera, arXiv:2412.03461.
  49. N. K. Johnson-McDaniel, S. Kulkarni, and A. Gupta, Phys. Rev. D 106, 023001 (2022).
  50. M. Mould and D. Gerosa, Phys. Rev. D 105, 024076 (2022).
  51. D. Gerosa, G. Fumagalli, M. Mould, G. Cavallotto, D. P. Monroy, D. Gangardt, and V. De Renzis, Phys. Rev. D 108, 024042 (2023).
  52. V. De Renzis, D. Gerosa, M. Mould, R. Buscicchio, and L. Zanga, Phys. Rev. D 108, 024024 (2023).
  53. S. Kulkarni, N. K. Johnson-McDaniel, K. S. Phukon, N. V. Krishnendu, and A. Gupta, Phys. Rev. D 109, 043002 (2024).
  54. G. Fumagalli and D. Gerosa, Phys. Rev. D 108, 124055 (2023).
  55. G. Fumagalli, I. Romero-Shaw, D. Gerosa, V. De Renzis, K. Kritos, and A. Olejak, Phys. Rev. D 110, 063012 (2024).
  56. L. Reali, M. Mould, D. Gerosa, and V. Varma, Classical Quantum Gravity 37, 225005 (2020).
  57. É. Racine, Phys. Rev. D 78, 044021 (2008).
  58. K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Phys. Rev. D 77, 064034 (2008).
  59. K. G. Arun, L. Blanchet, B. R. Iyer, and S. Sinha, Phys. Rev. D 80, 124018 (2009).
  60. A. Klein and P. Jetzer, Phys. Rev. D 81, 124001 (2010).
  61. A. Klein, Y. Boetzel, A. Gopakumar, P. Jetzer, and L. de Vittori, Phys. Rev. D 98, 104043 (2018).
  62. N. Loutrel and N. Yunes, Classical Quantum Gravity 34, 044003 (2017).
  63. LVK Algorithm Library Suite (lalsuite), 10.7935/GT1W-FZ16.
  64. eccentric_bbh_evolution_lalsuite git repository, https://gitlab.com/Amiteshh/eccentric_bbh_evolution_lalsuite/.
  65. K. S. Phukon, A. Gupta, S. Bose, and P. Jain, Phys. Rev. D 100, 124008 (2019).
  66. J. D. Schnittman, Phys. Rev. D 70, 124020 (2004).
  67. N. J. Cornish and J. S. Key, Phys. Rev. D 82, 044028 (2010); 84, 029901(E) (2011).
  68. P. Csizmadia, G. Debreczeni, I. Rácz, and M. Vasúth, Classical Quantum Gravity 29, 245002 (2012).
  69. B. Ireland, O. Birnholtz, H. Nakano, E. West, and M. Campanelli, Phys. Rev. D 100, 024015 (2019).
  70. A. Klein, arXiv:2106.10291.
  71. J. N. Arredondo, A. Klein, and N. Yunes, Phys. Rev. D 110, 044044 (2024).
  72. R. Sturani, Note on the derivation of the angular momentum and spin precessing equations in SpinTaylor codes, Technical Report No. LIGO-T1500554, LIGO Project, 2021, https://dcc.ligo.org/T1500554/public.
  73. A. Buonanno, Y.-b. Chen, and M. Vallisneri, Phys. Rev. D 67, 104025 (2003); 74, 029904(E) (2006).
  74. A. Nagar and P. Rettegno, Phys. Rev. D 104, 104004 (2021).
  75. A. Gamboa et al., arXiv:2412.12823.
  76. M. de Lluc Planas, A. Ramos-Buades, C. García-Quirós, H. Estellés, S. Husa, and M. Haney, arXiv:2503.13062.
  77. M. Boyle, R. Owen, and H. P. Pfeiffer, Phys. Rev. D 84, 124011 (2011).
  78. R. O’Shaughnessy, B. Vaishnav, J. Healy, Z. Meeks, and D. Shoemaker, Phys. Rev. D 84, 124002 (2011).
  79. P. Schmidt, M. Hannam, S. Husa, and P. Ajith, Phys. Rev. D 84, 024046 (2011).
  80. P. Schmidt, M. Hannam, and S. Husa, Phys. Rev. D 86, 104063 (2012).
  81. M. Kesden, D. Gerosa, R. O’Shaughnessy, E. Berti, and U. Sperhake, Phys. Rev. Lett. 114, 081103 (2015).
  82. D. Gerosa, M. Kesden, U. Sperhake, E. Berti, and R. O’Shaughnessy, Phys. Rev. D 92, 064016 (2015).
  83. D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics (Elsevier, New York, 2013).
  84. R.-M. Memmesheimer, A. Gopakumar, and G. Schäfer, Phys. Rev. D 70, 104011 (2004).
  85. T. Damour and N. Deruelle, Ann. Inst. Henri Poincaré Phys. Théor. 43, 107 (1985), https://www.numdam.org/item/AIHPA_1985__43_1_107_0/.
  86. T. Damour and N. Deruelle, Ann. Inst. Henri Poincaré Phys. Théor. 44, 263 (1986), https://www.numdam.org/item/AIHPA_1986__44_3_263_0/.
  87. C. Königsdörffer and A. Gopakumar, Phys. Rev. D 73, 124012 (2006).
  88. K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Phys. Rev. D 77, 064035 (2008).
  89. N. Loutrel, S. Liebersbach, N. Yunes, and N. Cornish, Classical Quantum Gravity 36, 025004 (2019).
  90. M. A. Shaikh, V. Varma, H. P. Pfeiffer, A. Ramos-Buades, and M. van de Meent, Phys. Rev. D 108, 104007 (2023).
  91. M. Boschini, N. Loutrel, D. Gerosa, and G. Fumagalli, Phys. Rev. D 111, 024008 (2025).
  92. T. Islam and T. Venumadhav, arXiv:2502.02739.
  93. T. Damour, A. Gopakumar, and B. R. Iyer, Phys. Rev. D 70, 064028 (2004).
  94. G. Cho, S. Tanay, A. Gopakumar, and H. M. Lee, Phys. Rev. D 105, 064010 (2022).
  95. M. Tessmer, J. Hartung, and G. Schäfer, Classical Quantum Gravity 30, 015007 (2013).
  96. Q. Henry and M. Khalil, Phys. Rev. D 108, 104016 (2023).
  97. L. E. Kidder, Phys. Rev. D 52, 821 (1995).
  98. T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Phys. Rev. D 49, 6274 (1994).
  99. L. Blanchet, Living Rev. Relativity 27, 4 (2024).
  100. K. Kritos, V. Strokov, V. Baibhav, and E. Berti, Phys. Rev. D 110, 043023 (2024).
  101. A. Bohé, S. Marsat, G. Faye, and L. Blanchet, Classical Quantum Gravity 30, 075017 (2013).
  102. A. Bohé, S. Marsat, and L. Blanchet, Classical Quantum Gravity 30, 135009 (2013).
  103. A. Bohé, G. Faye, S. Marsat, and E. K. Porter, Classical Quantum Gravity 32, 195010 (2015).
  104. R. Samanta, S. Tanay, and L. C. Stein, Phys. Rev. D 108, 124039 (2023).
  105. G. Fumagalli, N. Loutrel, D. Gerosa, and M. Boschini, Phys. Rev. D 112, 024012 (2025).
  106. C. Königsdörffer and A. Gopakumar, Phys. Rev. D 71, 024039 (2005).
  107. D. Gerosa and M. Kesden, Phys. Rev. D 93, 124066 (2016).
  108. A. Buonanno, L. E. Kidder, A. H. Mroue, H. P. Pfeiffer, and A. Taracchini, Phys. Rev. D 83, 104034 (2011).
  109. C. Cutler, D. Kennefick, and E. Poisson, Phys. Rev. D 50, 3816 (1994).
  110. V. Cardoso, C. F. B. Macedo, and R. Vicente, Phys. Rev. D 103, 023015 (2021).
  111. C. W. Lincoln and C. M. Will, Phys. Rev. D 42, 1123 (1990).
  112. N. Loutrel, S. Liebersbach, N. Yunes, and N. Cornish, Classical Quantum Gravity 36, 01LT01 (2019).
  113. C. M. Will, Classical Quantum Gravity 36, 195013 (2019).
  114. N. K. Johnson-McDaniel, K. S. Phukon, N. V. Krishnendu, and A. Gupta, Phys. Rev. D 108, 103003 (2023); 109, 129903(E) (2024).
  115. K. Wette, SoftwareX 12, 100634 (2020).
  116. A. Singh, N. K. Johnson-McDaniel, A. Gupta, and K. S. Phukon, arXiv:2505.07238.
  117. H. Yu, S. Ma, M. Giesler, and Y. Chen, Phys. Rev. D 102, 123009 (2020).
  118. I. Harry, S. Privitera, A. Bohé, and A. Buonanno, Phys. Rev. D 94, 024012 (2016).
  119. A. Chattaraj, T. RoyChowdhury, Divyajyoti, C. K. Mishra, and A. Gupta, Phys. Rev. D 106, 124008 (2022).
  120. D. Sun, M. Boyle, K. Mitman, M. A. Scheel, L. C. Stein, S. A. Teukolsky, and V. Varma, Phys. Rev. D 110, 104076 (2024).
  121. J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007).
  122. C. R. Harris et al., Nature (London) 585, 357 (2020).
  123. The Pandas Development Team, pandas software (2024).
  124. W. McKinney, in Proceedings of the 9th python in Science Conference, edited by S. van der Walt and J. Millman (2010), pp. 51–56.
  125. P. Virtanen et al., Nat. Methods 17, 261 (2020).
  126. K. S. Phukon, N. K. Johnson-McDaniel, A. Singh, and A. Gupta, Data release: Evolution of precessing binary black holes on eccentric orbits using orbit-averaged evolution equations, Zenodo, 10.5281/zenodo.16881720.
  127. L. Á. Gergely and Z. Keresztes, Phys. Rev. D 67, 024020 (2003).
  128. A. Klein (private communication).
  129. lalsimulation XLALSimInspiralSpinTaylorPNEvolveOrbit function documentation, https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_inspiral_spin_taylor__c.html#ga35cfdf3082e09cc97cda9e11ba4c2bff.

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Phys. Rev. D 112, 044067– Published 29 August, 2025
  • Received 2 May 2025
  • Accepted 3 August 2025
  • Corrected 14 January 2026
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