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Sixth post-Newtonian nonlocal-in-time dynamics of binary systems

Donato Bini, Thibault Damour, Andrea Geralico

TL;DR

This work advances the conservative two-body dynamics of General Relativity to the sixth post-Newtonian level by fully incorporating nonlocal-in-time tail effects. Using a two-part Hamiltonian split with a flexible time rescaling $f(t)$, the authors derive gauge-invariant characterizations of the nonlocal dynamics: the nonlocal tail contribution to the hyperbolic-scattering angle and the Delaunay-averaged nonlocal Hamiltonian for elliptic orbits. They provide explicit 6PN expressions for the h-route tail in both large-$J$ and small-$e$ limits, reveal the appearance of $\zeta(3)$, and analyze the mass-ratio dependence of the GW energy loss. The work also determines a minimal, gauge-fixed choice for the flexibility factor and translates nonlocal tail contributions into an EOB framework, including a nonlocal $Q$-potential; it further yields circular-orbit observables such as the nonlocal energy and periastron precession, thereby enhancing the connection between PN, PM, EFT, and self-force approaches.

Abstract

We complete our previous derivation, at the sixth post-Newtonian (6PN) accuracy, of the local-in-time dynamics of a gravitationally interacting two-body system by giving two gauge-invariant characterizations of its complementary nonlocal-in-time dynamics. On the one hand, we compute the nonlocal part of the scattering angle for hyberboliclike motions; and, on the other hand, we compute the nonlocal part of the averaged (Delaunay) Hamiltonian for ellipticlike motions. The former is computed as a large-angular-momentum expansion (given here to next-to-next-to-leading order), while the latter is given as a small-eccentricity expansion (given here to the tenth order). We note the appearance of $ζ(3)$ in the nonlocal part of the scattering angle. The averaged Hamiltonian for ellipticlike motions then yields two more gauge-invariant observables: the energy and the periastron precession as functions of orbital frequencies. We point out the existence of a hidden simplicity in the mass-ratio dependence of the gravitational-wave energy loss of a two-body system.

Sixth post-Newtonian nonlocal-in-time dynamics of binary systems

TL;DR

This work advances the conservative two-body dynamics of General Relativity to the sixth post-Newtonian level by fully incorporating nonlocal-in-time tail effects. Using a two-part Hamiltonian split with a flexible time rescaling , the authors derive gauge-invariant characterizations of the nonlocal dynamics: the nonlocal tail contribution to the hyperbolic-scattering angle and the Delaunay-averaged nonlocal Hamiltonian for elliptic orbits. They provide explicit 6PN expressions for the h-route tail in both large- and small- limits, reveal the appearance of , and analyze the mass-ratio dependence of the GW energy loss. The work also determines a minimal, gauge-fixed choice for the flexibility factor and translates nonlocal tail contributions into an EOB framework, including a nonlocal -potential; it further yields circular-orbit observables such as the nonlocal energy and periastron precession, thereby enhancing the connection between PN, PM, EFT, and self-force approaches.

Abstract

We complete our previous derivation, at the sixth post-Newtonian (6PN) accuracy, of the local-in-time dynamics of a gravitationally interacting two-body system by giving two gauge-invariant characterizations of its complementary nonlocal-in-time dynamics. On the one hand, we compute the nonlocal part of the scattering angle for hyberboliclike motions; and, on the other hand, we compute the nonlocal part of the averaged (Delaunay) Hamiltonian for ellipticlike motions. The former is computed as a large-angular-momentum expansion (given here to next-to-next-to-leading order), while the latter is given as a small-eccentricity expansion (given here to the tenth order). We note the appearance of in the nonlocal part of the scattering angle. The averaged Hamiltonian for ellipticlike motions then yields two more gauge-invariant observables: the energy and the periastron precession as functions of orbital frequencies. We point out the existence of a hidden simplicity in the mass-ratio dependence of the gravitational-wave energy loss of a two-body system.

Paper Structure

This paper contains 31 sections, 362 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Brief reminder about the nonlocal part of the action
  3. Scattering angle
  4. Computation of $W^{\rm tail,h}\equiv \langle H_{\rm nonloc,h}^{4+5+6 \rm PN} \rangle_\infty$
  5. The 2PN-accurate n-polar moments
  6. The harmonic-coordinate quasi-Keplerian parametrization of the hyperbolic motion
  7. $W^{\rm tail,h}$ along hyperbolic orbits: computational details
  8. Value of $W^{\rm tail, h}$ up to the NNLO in $e_r^{-1}$ and at the fractional 2PN accuracy
  9. Computation of $W^{\rm tail,h}(E,j)$ in the frequency domain
  10. Final results for the h-route tail contribution to the scattering angle
  11. Second-order tail contribution to the integrated action and to the scattering angle: $W^{5.5{\rm PN}}$ and $\chi^{5.5{\rm PN}}$
  12. Analysis of the $\nu$-dependence of the harmonic-coordinate nonlocal scattering angle $\chi^{\rm nonloc,h}$
  13. Reminder of the $\nu$-rule to be satisfied
  14. On the $\nu$-structure of the logarithmic contributions, and of the gravitational-wave energy loss
  15. Contributions to $\widetilde{\chi}_n^{\rm nonloc,h}$ violating the special $\nu$-structure
  16. ...and 16 more sections