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Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders

Donato Bini, Thibault Damour, Andrea Geralico

TL;DR

The paper advances conservative two-body dynamics to $5$PN and its $5.5$PN tail, with a $6$PN extension in the $3$PM sector, by marrying PN, PM, SF, EOB and Delaunay methods. It builds a nonlocal tail action and its local EOB Hamiltonian, fixes the linear-in-$ u$ SF contributions, and then, via a canonical transformation to an energy gauge, derives a complete gauge-fixed local Hamiltonian at $5$PN with two remaining nonlinear-in-$ u$ coefficients to be determined. The authors confirm the $3$PM results of Bern et al. at $6$PN, compute explicit scattering-angle expansions up to $6$PN, and provide a thorough, cross-validated EOB description, including a tail$^2$ contribution at $5.5$PN and a predictive $q_{8,2.5}$ term for future SF checks. A striking outcome is the logarithm-free nature of the $5$PN local sector and the simple, gauge-invariant structure of the radial action alongside the scattering angle, highlighting a hidden simplicity in the high-order binary dynamics with wide implications for analytic gravitational-wave modeling.

Abstract

Using the new methodology introduced in a recent Letter [Phys.\ Rev.\ Lett.\ {\bf 123}, 231104 (2019)], we present the details of the computation of the conservative dynamics of gravitationally interacting binary systems at the fifth post-Newtonian (5PN) level, together with its extension at the fifth-and-a-half post-Newtonian (5.5PN) level. We present also the sixth post-Newtonian (6PN) contribution to the third-post-Minkowskian (3PM) dynamics. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, effective one-body, and Delaunay averaging. We determine the full functional structure of the 5PN Hamiltonian (which involves 95 non-zero numerical coefficients), except for two undetermined coefficients proportional to the cube of the symmetric mass ratio, and to the fifth and sixth power of the gravitational constant, $G$. We present not only the 5PN-accurate, 3PM contribution to the scattering angle, but also its 6PN-accurate generalization. Both results agree with the corresponding truncations of the recent 3PM result of Bern et al. [Phys.\ Rev.\ Lett.\ {\bf 122}, 201603 (2019)]. We also compute the 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scattering angle, including its nonlocal contribution, thereby offering checks for future 4PM calculations. We point out a remarkable hidden simplicity of the gauge-invariant functional relation between the radial action and the effective-one-body energy and angular momentum.

Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders

TL;DR

The paper advances conservative two-body dynamics to PN and its PN tail, with a PN extension in the PM sector, by marrying PN, PM, SF, EOB and Delaunay methods. It builds a nonlocal tail action and its local EOB Hamiltonian, fixes the linear-in- SF contributions, and then, via a canonical transformation to an energy gauge, derives a complete gauge-fixed local Hamiltonian at PN with two remaining nonlinear-in- coefficients to be determined. The authors confirm the PM results of Bern et al. at PN, compute explicit scattering-angle expansions up to PN, and provide a thorough, cross-validated EOB description, including a tail contribution at PN and a predictive term for future SF checks. A striking outcome is the logarithm-free nature of the PN local sector and the simple, gauge-invariant structure of the radial action alongside the scattering angle, highlighting a hidden simplicity in the high-order binary dynamics with wide implications for analytic gravitational-wave modeling.

Abstract

Using the new methodology introduced in a recent Letter [Phys.\ Rev.\ Lett.\ {\bf 123}, 231104 (2019)], we present the details of the computation of the conservative dynamics of gravitationally interacting binary systems at the fifth post-Newtonian (5PN) level, together with its extension at the fifth-and-a-half post-Newtonian (5.5PN) level. We present also the sixth post-Newtonian (6PN) contribution to the third-post-Minkowskian (3PM) dynamics. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, effective one-body, and Delaunay averaging. We determine the full functional structure of the 5PN Hamiltonian (which involves 95 non-zero numerical coefficients), except for two undetermined coefficients proportional to the cube of the symmetric mass ratio, and to the fifth and sixth power of the gravitational constant, . We present not only the 5PN-accurate, 3PM contribution to the scattering angle, but also its 6PN-accurate generalization. Both results agree with the corresponding truncations of the recent 3PM result of Bern et al. [Phys.\ Rev.\ Lett.\ {\bf 122}, 201603 (2019)]. We also compute the 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scattering angle, including its nonlocal contribution, thereby offering checks for future 4PM calculations. We point out a remarkable hidden simplicity of the gauge-invariant functional relation between the radial action and the effective-one-body energy and angular momentum.

Paper Structure

This paper contains 21 sections, 221 equations, 2 figures, 14 tables.

Table of Contents

  1. Introduction
  2. The 5PN-accurate nonlocal action and its associated Hamiltonian
  3. Using self-force theory to compute the local-plus-nonlocal EOB Hamiltonian in standard gauge, at first order in $\nu$
  4. Obtaining the 5PN-accurate local EOB Hamiltonian at linear order in $\nu$
  5. The 5PN local EOB Hamiltonian: from the standard $p_r$-gauge to the energy-gauge
  6. Determination of the 3PM dynamics up to the 5PN (and 6PN) levels
  7. Nonlocal part of the scattering angle
  8. Use and determination of the flexibility factor $f(t)$
  9. Using the mass-ratio dependence of the scattering angle to determine most of the $\nu^{n\geq 2}$ structure of the $f$-route local Hamiltonian
  10. Values of the 5PN-accurate $f$- route local scattering angle at PM orders $G^3$, $G^4$, $G^5$ and $G^6$
  11. Final results for the 5PN-accurate $f$-route local EOB Hamiltonian
  12. 5PN-accurate $f$-flexed local effective EOB Hamiltonian in energy-gauge
  13. 5PN-accurate $f$-type local effective EOB Hamiltonian in $p_r$-gauge
  14. The standard ($f$-type local) EOB Hamiltonian at 5PN
  15. Canonical transformation between the $p_r$-gauge and the energy-gauge
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Matrix of the 97 non-zero numerical coefficients $C^{(2 l)}_{2k, n}$ encoding the various powers of $\nu$ in the Hamiltonian \ref{['Heobfin']}.
  • Figure 2: Schematic representation of the theoretical tools used to obtain the various contributions to the 5PN-accurate local Hamiltonian, adapted from Ref. Bini:2019nra. These contributions are keyed, on the horizontal axis, by powers of $u=GM/r$ and squared momentum $p^2 \sim p_r^2$, and, on the the vertical axis, by powers of $\nu \equiv m_1 m_2/(m_1+m_2)^2$. The checks indicate the coefficients determined in the present work. The question marks denote the only two missing coefficients. Note that even if certain coefficients in Table \ref{['table_Hreal']} only include terms up to $O(\nu^5)$, the identification $p^2 \sim p_r^2$ done in this schematic figure lumps terms together so that $O(\nu^6)$ terms arise in each column.