- 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

Revisiting the coprecessing frame in the presence of orbital eccentricity

Lucy M. Thomas1,2,*, Katerina Chatziioannou1,2,†, Sam Johar3,2, Taylor Knapp1,2,‡, and Michael Boyle4

  • *Contact author: lmthomas@caltech.edu
  • †Contact author: kchatziioannou@caltech.edu
  • ‡Contact author: tknapp@caltech.edu
PDF

Phys. Rev. D 114, 084090 – Published 28 October, 2026

DOI: https://doi.org/10.1103/zv2n-76n3

Abstract

Accurate inclusion of both spin precession and orbital eccentricity effects in gravitational waveform models represents a key hurdle in our ability to fully characterize the properties of compact binaries. Virtually all efforts to model precession rely on a coprecessing frame transformation, a time-dependent spatial rotation that tracks the dominant emission direction and simplifies the waveform morphology. We assess the utility of the coprecessing frame transformation to separate out the effect of the precession of the orbital plane from the waveform in the presence of non-negligible orbital eccentricity. We rely on 20 numerical relativity simulations, which include the complete physical effects of spin precession and eccentricity in the strong field, and compare waveforms in both the inertial and coprecessing frames. Comparing against the eccentric, spin-aligned model SEOBNRv5EHM, we find that while the waveform mismatches decrease in the coprecessing frame, they remain above the level required for accurate waveform modeling, ∼0.01 or higher for large inclinations. Further improvements, e.g., modeling mode asymmetries as already pursued for quasicircular binaries, will likely prove essential. We also find that by removing the dominant amplitude and phase modulations from the waveform, the coprecessing frame facilitates surrogate modeling, achieving lower errors at a fixed number of basis elements compared to the inertial frame. Our results demonstrate both the utility and the limitations of the coprecessing frame as a cornerstone in waveform modeling for eccentric and precessing binaries.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (94)

  1. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the Fourth LIGO-Virgo-KAGRA Observing Run, Astrophys. J. Lett. 1004, L22 (2026).
  2. Digvijay Wadekar, Javier Roulet, Tejaswi Venumadhav, Ajit Kumar Mehta, Barak Zackay, Jonathan Mushkin, Seth Olsen, and Matias Zaldarriaga, New black hole mergers in the LIGO-Virgo O3 data from a gravitational wave search including higher-order harmonics, arXiv:2312.06631.
  3. Alexander H. Nitz, Sumit Kumar, Yi-Fan Wang, Shilpa Kastha, Shichao Wu, Marlin Schäfer, Rahul Dhurkunde, and Collin D. Capano, 4-OGC: Catalog of gravitational waves from compact binary mergers, Astrophys. J. 946, 59 (2023).
  4. P. C. Peters and J. Mathews, Gravitational radiation from point masses in a Keplerian orbit, Phys. Rev. 131, 435 (1963).
  5. P. C. Peters, Gravitational radiation and the motion of two point masses, Phys. Rev. 136, B1224 (1964).
  6. Marcus E. Lower, Eric Thrane, Paul D. Lasky, and Rory Smith, Measuring eccentricity in binary black hole inspirals with gravitational waves, Phys. Rev. D 98, 083028 (2018).
  7. Michael Zevin, Isobel M. Romero-Shaw, Kyle Kremer, Eric Thrane, and Paul D. Lasky, Implications of eccentric observations on binary black hole formation channels, Astrophys. J. Lett. 921, L43 (2021).
  8. V. Gayathri, J. Healy, J. Lange, B. O’Brien, M. Szczepanczyk, Imre Bartos, M. Campanelli, S. Klimenko, C. O. Lousto, and R. O’Shaughnessy, Eccentricity estimate for black hole mergers with numerical relativity simulations, Nat. Astron. 6, 344 (2022).
  9. Rossella Gamba, Matteo Breschi, Gregorio Carullo, Simone Albanesi, Piero Rettegno, Sebastiano Bernuzzi, and Alessandro Nagar, GW190521 as a dynamical capture of two nonspinning black holes, Nat. Astron. 7, 11 (2023).
  10. H. L. Iglesias et al., Eccentricity estimation for five binary black hole mergers with higher-order gravitational-wave modes, Astrophys. J. 972, 65 (2024).
  11. Nihar Gupte et al., Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA, Phys. Rev. D 112, 104045 (2025).
  12. Maria de Lluc Planas, Antoni Ramos-Buades, Cecilio García-Quirós, Héctor Estellés, Sascha Husa, and Maria Haney, Reanalysis of binary black hole gravitational wave events for orbital eccentricity signatures, Phys. Rev. D 112, 123004 (2025).
  13. Isobel Romero-Shaw, Jakob Stegmann, Hiromichi Tagawa, Davide Gerosa, Johan Samsing, Nihar Gupte, and Stephen R. Green, GW200208_222617 as an eccentric black-hole binary merger: Properties and astrophysical implications, Phys. Rev. D 112, 063052 (2025).
  14. Gonzalo Morras, Geraint Pratten, and Patricia Schmidt, Orbital eccentricity in a neutron star—black hole binary merger (2025).
  15. Aasim Jan, Bing-Jyun Tsao, Richard O’Shaughnessy, Deirdre Shoemaker, and Pablo Laguna, GW200105: A detailed study of eccentricity in the neutron star-black hole binary, Phys. Rev. D 113, 024018 (2026).
  16. Theocharis A. Apostolatos, Curt Cutler, Gerald J. Sussman, and Kip S. Thorne, Spin induced orbital precession and its modulation of the gravitational wave forms from merging binaries, Phys. Rev. D 49, 6274 (1994).
  17. Antoni Ramos-Buades, Sascha Husa, Geraint Pratten, Héctor Estellés, Cecilio García-Quirós, Maite Mateu-Lucena, Marta Colleoni, and Rafel Jaume, First survey of spinning eccentric black hole mergers: Numerical relativity simulations, hybrid waveforms, and parameter estimation, Phys. Rev. D 101, 083015 (2020).
  18. Juan Calderón Bustillo, Nicolas Sanchis-Gual, Alejandro Torres-Forné, and José A. Font, Confusing head-on collisions with precessing intermediate-mass binary black hole mergers, Phys. Rev. Lett. 126, 201101 (2021).
  19. Snehal Tibrewal, Aaron Zimmerman, Jacob Lange, and Deirdre Shoemaker, Misinterpreting spin precession as orbital eccentricity in gravitational-wave signals, Phys. Rev. D 114, 064062 (2026).
  20. Tamal RoyChowdhury, V. Gayathri, Rossella Gamba, Shubhagata Bhaumik, Imre Bartos, and Jolien Creighton, The cost of circularity: Quantifying eccentricity-induced biases in binary black hole inference, arXiv:2603.02453.
  21. Isobel M. Romero-Shaw, Paul D. Lasky, Eric Thrane, and Juan Calderon Bustillo, GW190521: Orbital eccentricity and signatures of dynamical formation in a binary black hole merger signal, Astrophys. J. Lett. 903, L5 (2020).
  22. Isobel M. Romero-Shaw, Davide Gerosa, and Nicholas Loutrel, Eccentricity or spin precession? Distinguishing subdominant effects in gravitational-wave data, Mon. Not. R. Astron. Soc. 519, 5352 (2023).
  23. Gonzalo Morras, Geraint Pratten, and Patricia Schmidt, Improved post-Newtonian waveform model for inspiralling precessing-eccentric compact binaries, Phys. Rev. D 111, 084052 (2025).
  24. Laszlo A. Gergely, Zoltan I. Perjes, and Matyas Vasuth, Spin effects in gravitational radiation back reaction. 3. Compact binaries with two spinning components, Phys. Rev. D 58, 124001 (1998).
  25. Raoul-Martin Memmesheimer, Achamveedu Gopakumar, and Gerhard Schaefer, Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits, Phys. Rev. D 70, 104011 (2004).
  26. Christian Konigsdorffer and Achamveedu Gopakumar, Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The Leading order spin-orbit interaction, Phys. Rev. D 71, 024039 (2005).
  27. Antoine Klein and Philippe Jetzer, Spin effects in the phasing of gravitational waves from binaries on eccentric orbits, Phys. Rev. D 81, 124001 (2010).
  28. Neil J. Cornish and Joey Shapiro Key, Computing waveforms for spinning compact binaries in quasi-eccentric orbits, Phys. Rev. D 82, 044028 (2010); 84, 029901(E) (2011).
  29. Janna Levin, Sean T. McWilliams, and Hugo Contreras, Inspiral of generic black hole binaries: Spin, precession, and eccentricity, Classical Quantum Gravity 28, 175001 (2011).
  30. Achamveedu Gopakumar and Gerhard Schafer, Gravitational wave phasing for spinning compact binaries in inspiraling eccentric orbits, Phys. Rev. D 84, 124007 (2011).
  31. Manuel Tessmer and Gerhard Schäfer, Eccentric motion of spinning compact binaries, Phys. Rev. D 89, 104055 (2014).
  32. Giulia Fumagalli and Davide Gerosa, Spin-eccentricity interplay in merging binary black holes, Phys. Rev. D 108, 124055 (2023).
  33. Quentin Henry and Mohammed Khalil, Spin effects in gravitational waveforms and fluxes for binaries on eccentric orbits to the third post-Newtonian order, Phys. Rev. D 108, 104016 (2023).
  34. Khun Sang Phukon, Nathan K. Johnson-McDaniel, Amitesh Singh, and Anuradha Gupta, Evolution of precessing binary black holes on eccentric orbits using orbit-averaged evolution equations, Phys. Rev. D 112, 044067 (2025).
  35. Rossella Gamba, Danilo Chiaramello, and Sayan Neogi, Toward efficient effective-one-body models for generic, nonplanar orbits, Phys. Rev. D 110, 024031 (2024).
  36. Aldo Gamboa et al., Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model seobnrv5ehm, Phys. Rev. D 112, 044038 (2025).
  37. Alessandro Nagar, Danilo Chiaramello, Rossella Gamba, Simone Albanesi, Sebastiano Bernuzzi, Veronica Fantini, Mattia Panzeri, and Piero Rettegno, Effective-one-body waveform model for noncircularized, planar, coalescing black hole binaries. II. High accuracy by improving logarithmic terms in resummations, Phys. Rev. D 111, 064050 (2025).
  38. Kaushik Paul, Akash Maurya, Quentin Henry, Kartikey Sharma, Pranav Satheesh, Divyajyoti, Prayush Kumar, and Chandra Kant Mishra, Eccentric, spinning, inspiral-merger-ringdown waveform model with higher modes for the detection and characterization of binary black holes, Phys. Rev. D 111, 084074 (2025).
  39. Antoni Ramos-Buades, Quentin Henry, and Maria Haney, Fast frequency-domain phenomenological modeling of eccentric aligned-spin binary black holes, arXiv:2601.03340.
  40. Mark A. Scheel et al., The SXS collaboration’s third catalog of binary black hole simulations, Classical Quantum Gravity 42, 195017 (2025).
  41. Deborah Ferguson et al., Second MAYA catalog of binary black hole numerical relativity waveforms, Phys. Rev. D 112, 044043 (2025).
  42. James Healy and Carlos O. Lousto, Third RIT binary black hole simulations catalog, Phys. Rev. D 102, 104018 (2020).
  43. Aditya Vijaykumar, Alexandra G. Hanselman, and Michael Zevin, Consistent eccentricities for gravitational-wave astronomy: Resolving discrepancies between astrophysical simulations and waveform models, Astrophys. J. 969, 132 (2024).
  44. Xiaolin Liu, Zhoujian Cao, and Zong-Hong Zhu, Effective-one-body numerical-relativity waveform model for eccentric spin-precessing binary black hole coalescence, Classical Quantum Gravity 41, 195019 (2024).
  45. Simone Albanesi, Rossella Gamba, Sebastiano Bernuzzi, Joan Fontbuté, Alejandra Gonzalez, and Alessandro Nagar, Effective-one-body modeling for generic compact binaries with arbitrary orbits, arXiv:2503.14580.
  46. Patricia Schmidt, Mark Hannam, and Sascha Husa, Towards models of gravitational waveforms from generic binaries: A simple approximate mapping between precessing and non-precessing inspiral signals, Phys. Rev. D 86, 104063 (2012).
  47. R. O’Shaughnessy, J. Healy, L. London, Z. Meeks, and D. Shoemaker, Is J enough? Comparison of gravitational waves emitted along the total angular momentum direction with other preferred orientations, Phys. Rev. D 85, 084003 (2012); 87, 069901(E) (2013).
  48. Eleanor Hamilton et al., PhenomXPNR: An improved gravitational wave model linking precessing inspirals and NR-calibrated merger-ringdown, arXiv:2507.02604.
  49. Yi Pan, Alessandra Buonanno, Andrea Taracchini, Lawrence E. Kidder, Abdul H. Mroué, Harald P. Pfeiffer, Mark A. Scheel, and Béla Szilágyi, Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism, Phys. Rev. D 89, 084006 (2014).
  50. Rossella Gamba, Sarp Akçay, Sebastiano Bernuzzi, and Jake Williams, Effective-one-body waveforms for precessing coalescing compact binaries with post-Newtonian twist, Phys. Rev. D 106, 024020 (2022).
  51. Antoni Ramos-Buades, Alessandra Buonanno, Héctor Estellés, Mohammed Khalil, Deyan P. Mihaylov, Serguei Ossokine, Lorenzo Pompili, and Mahlet Shiferaw, Next generation of accurate and efficient multipolar precessing-spin effective-one-body waveforms for binary black holes, Phys. Rev. D 108, 124037 (2023).
  52. Michael Boyle, Lawrence E. Kidder, Serguei Ossokine, and Harald P. Pfeiffer, Gravitational-wave modes from precessing black-hole binaries, arXiv:1409.4431.
  53. Jonathan E. Thompson, Eleanor Hamilton, Lionel London, Shrobana Ghosh, Panagiota Kolitsidou, Charlie Hoy, and Mark Hannam, PhenomXO4a: A phenomenological gravitational-wave model for precessing black-hole binaries with higher multipoles and asymmetries, Phys. Rev. D 109, 063012 (2024).
  54. Héctor Estellés, Alessandra Buonanno, Raffi Enficiaud, Cheng Foo, and Lorenzo Pompili, Adding equatorial-asymmetric effects for spin-precessing binaries into the SEOBNRv5PHM waveform model, Phys. Rev. D 113, 044049 (2026).
  55. Eleanor Hamilton, Lionel London, and Mark Hannam, Ringdown frequencies in black holes formed from precessing black-hole binaries, Phys. Rev. D 107, 104035 (2023).
  56. Md Arif Shaikh, Vijay Varma, Antoni Ramos-Buades, Harald P. Pfeiffer, Michael Boyle, Lawrence E. Kidder, and Mark A. Scheel, Defining eccentricity for spin-precessing binaries, Classical Quantum Gravity 42, 195012 (2025).
  57. Lucy Thomas, Eccentric precessing mismatch data repository, 2026.
  58. Michael Boyle, scri:Python/numba code for manipulating time-dependent functions of spin-weighted spherical harmonics on future null infinity.
  59. Michael Boyle, Transformations of asymptotic gravitational-wave data, Phys. Rev. D 93, 084031 (2016).
  60. Lawrence E. Kidder, Coalescing binary systems of compact objects to postNewtonian 5/2 order. 5. Spin effects, Phys. Rev. D 52, 821 (1995).
  61. Eliot Finch and Christopher J. Moore, Modeling the ringdown from precessing black hole binaries, Phys. Rev. D 103, 084048 (2021).
  62. Hengrui Zhu et al., Black hole spectroscopy for precessing binary black hole coalescences, Phys. Rev. D 111, 064052 (2025).
  63. Alessandra Buonanno, Lawrence E. Kidder, and Luis Lehner, Estimating the final spin of a binary black hole coalescence, Phys. Rev. D 77, 026004 (2008).
  64. Latham Boyle and Michael Kesden, The spin expansion for binary black hole merger: New predictions and future directions, Phys. Rev. D 78, 024017 (2008).
  65. Luciano Rezzolla, Enrico Barausse, Ernst Nils Dorband, Denis Pollney, Christian Reisswig, Jennifer Seiler, and Sascha Husa, On the final spin from the coalescence of two black holes, Phys. Rev. D 78, 044002 (2008).
  66. Manuela Campanelli, Carlos O. Lousto, Yosef Zlochower, and David Merritt, Large merger recoils and spin flips from generic black-hole binaries, Astrophys. J. Lett. 659, L5 (2007).
  67. Carlos O. Lousto and Yosef Zlochower, Nonlinear gravitational recoil from the mergers of precessing black-hole binaries, Phys. Rev. D 87, 084027 (2013).
  68. Yosef Zlochower and Carlos O. Lousto, Modeling the remnant mass, spin, and recoil from unequal-mass, precessing black-hole binaries: The intermediate mass ratio regime, Phys. Rev. D 92, 024022 (2015); 94, 029901(E) (2016).
  69. Fabian Hofmann, Enrico Barausse, and Luciano Rezzolla, The final spin from binary black holes in quasi-circular orbits, Astrophys. J. Lett. 825, L19 (2016).
  70. Antoni Ramos-Buades, Patricia Schmidt, Geraint Pratten, and Sascha Husa, Validity of common modeling approximations for precessing binary black holes with higher-order modes, Phys. Rev. D 101, 103014 (2020).
  71. R. O’Shaughnessy, B. Vaishnav, J. Healy, Z. Meeks, and D. Shoemaker, Efficient asymptotic frame selection for binary black hole spacetimes using asymptotic radiation, Phys. Rev. D 84, 124002 (2011).
  72. Luc Blanchet, Gravitational radiation from postNewtonian sources and inspiraling compact binaries, Living Rev. Relativity 5, 3 (2002).
  73. Michael Boyle, Robert Owen, and Harald P. Pfeiffer, A geometric approach to the precession of compact binaries, Phys. Rev. D 84, 124011 (2011).
  74. Patricia Schmidt, Mark Hannam, Sascha Husa, and P. Ajith, Tracking the precession of compact binaries from their gravitational-wave signal, Phys. Rev. D 84, 024046 (2011).
  75. Alessandra Buonanno, Yan-bei Chen, and Michele Vallisneri, Detecting gravitational waves from precessing binaries of spinning compact objects: Adiabatic limit, Phys. Rev. D 67, 104025 (2003); 74, 029904(E) (2006).
  76. Md Arif Shaikh, Vijay Varma, Harald P. Pfeiffer, Antoni Ramos-Buades, and Maarten van de Meent, Defining eccentricity for gravitational wave astronomy, Phys. Rev. D 108, 104007 (2023).
  77. Collin Capano, Yi Pan, and Alessandra Buonanno, Impact of higher harmonics in searching for gravitational waves from nonspinning binary black holes, Phys. Rev. D 89, 102003 (2014).
  78. Jonathan Blackman, Scott E. Field, Mark A. Scheel, Chad R. Galley, Christian D. Ott, Michael Boyle, Lawrence E. Kidder, Harald P. Pfeiffer, and Béla Szilágyi, Numerical relativity waveform surrogate model for generically precessing binary black hole mergers, Phys. Rev. D 96, 024058 (2017).
  79. Katerina Chatziioannou, Antoine Klein, Nicolás Yunes, and Neil Cornish, Constructing gravitational waves from generic spin-precessing compact binary inspirals, Phys. Rev. D 95, 104004 (2017).
  80. Michael Pürrer and Carl-Johan Haster, Gravitational waveform accuracy requirements for future ground-based detectors, Phys. Rev. Res. 2, 023151 (2020).
  81. Alexandre Toubiana and Jonathan R. Gair, Indistinguishability criterion and estimating the presence of biases, arXiv:2401.06845.
  82. Jonathan E. Thompson, Charlie Hoy, Edward Fauchon-Jones, and Mark Hannam, Use and interpretation of signal-model indistinguishability measures for gravitational-wave astronomy, Phys. Rev. D 112, 064011 (2025).
  83. Alessandra Buonanno, Bala Iyer, Evan Ochsner, Yi Pan, and B. S. Sathyaprakash, Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors, Phys. Rev. D 80, 084043 (2009).
  84. Sophie Bini, Krzysztof Król, Katerina Chatziioannou, and Maximiliano Isi, The impact of waveform systematics and Gaussian noise on the interpretation of GW231123, Phys. Rev. D 113, 083036 (2026).
  85. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), GW231123: A binary black hole merger with total mass 190–265M⊙, Astrophys. J. Lett. 993, L25 (2025).
  86. Vijay Varma, Scott E. Field, Mark A. Scheel, Jonathan Blackman, Davide Gerosa, Leo C. Stein, Lawrence E. Kidder, and Harald P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses, Phys. Rev. Res. 1, 033015 (2019).
  87. Peter James Nee et al., Eccentric binary black holes: A new framework for numerical relativity waveform surrogates, arXiv:2510.00106.
  88. Akash Maurya, Prayush Kumar, Scott E. Field, Chandra Kant Mishra, Peter James Nee, Kaushik Paul, Harald P. Pfeiffer, Adhrit Ravichandran, and Vijay Varma, Chase orbits, not time: A scalable paradigm for long-duration eccentric gravitational-wave surrogates, Phys. Rev. Lett. 137, 041401 (2026).
  89. Michael Boyle, Keefe Mitman, Mark Scheel, and Leo Stein, The sxs package (v2025.0.15),.
  90. Deyan P. Mihaylov, Serguei Ossokine, Alessandra Buonanno, Hector Estelles, Lorenzo Pompili, Michael Pürrer, and Antoni Ramos-Buades, pyseobnr: A software package for the next generation of effective-one-body multipolar waveform models, SoftwareX 30, 102080 (2025).
  91. LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, LVK Algorithm Library—LALSuite, Free software (GPL), 2018.
  92. Karl Wette, SWIGLAL: Python and Octave interfaces to the LALSuite gravitational-wave data analysis libraries, SoftwareX 12, 100634 (2020).
  93. Pauli Virtanen et al. (scipy1.0 Contributors), scipy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
  94. Douglas Thain, Todd Tannenbaum, and Miron Livny, Distributed computing in practice: The condor experience, Concurrency Comput. Pract. Exper. 17, 323 (2005).

Outline

Information

Phys. Rev. D 114, 084090– Published 28 October, 2026
  • Received 1 April 2026
  • Accepted 16 July 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