Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Confirming and improving post-Newtonian and effective-one-body results from self-force computations along eccentric orbits around a Schwarzschild black hole

Donato Bini, Thibault Damour, Andrea Geralico

TL;DR

The paper computes high-order self-force corrections for eccentric orbits around a Schwarzschild black hole, delivering a 6.5PN accuracy for the $O(e^2)$ piece of the gauge-invariant redshift correction $\delta U$ and a 4PN accuracy for the $O(e^4)$ piece, with explicit mappings to the EOB radial potential $\bar{d}(u)$. It verifies these results against recent 4PN, 5PN, and 5.5PN predictions and against numerical SF data, providing the first independent analytic confirmation of the 4PN noncircular dynamics and reinforcing consistency across PN, SF, and EOB approaches. The work also analyzes the strong-field behavior near the last stable orbit, discusses the limitations of $O(e^2)$ expansions, and proposes using alternative gauge-invariant observables to probe the full strong-field regime. These results enhance the predictive power of the EOB model for eccentric binaries and guide future SF calculations to determine higher-order eccentric corrections and additional EOB potentials such as $q(u)$.

Abstract

We analytically compute, through the six-and-a-half post-Newtonian order, the second-order-in-eccentricity piece of the Detweiler-Barack-Sago gauge-invariant redshift function for a small mass in eccentric orbit around a Schwarzschild black hole. Using the first law of mechanics for eccentric orbits [A. Le Tiec, Phys. Rev. D {\bf 92}, 084021 (2015)] we transcribe our result into a correspondingly accurate knowledge of the second radial potential of the effective-one-body formalism [A. Buonanno and T. Damour, Phys. Rev. D {\bf 59}, 084006 (1999)]. We compare our newly acquired analytical information to several different numerical self-force data and find good agreement, within estimated error bars. We also obtain, for the first time, independent analytical checks of the recently derived, comparable-mass fourth-post-Newtonian order dynamics [T. Damour, P. Jaranowski and G. Shaefer, Phys. Rev. D {\bf 89}, 064058 (2014)].

Confirming and improving post-Newtonian and effective-one-body results from self-force computations along eccentric orbits around a Schwarzschild black hole

TL;DR

The paper computes high-order self-force corrections for eccentric orbits around a Schwarzschild black hole, delivering a 6.5PN accuracy for the piece of the gauge-invariant redshift correction and a 4PN accuracy for the piece, with explicit mappings to the EOB radial potential . It verifies these results against recent 4PN, 5PN, and 5.5PN predictions and against numerical SF data, providing the first independent analytic confirmation of the 4PN noncircular dynamics and reinforcing consistency across PN, SF, and EOB approaches. The work also analyzes the strong-field behavior near the last stable orbit, discusses the limitations of expansions, and proposes using alternative gauge-invariant observables to probe the full strong-field regime. These results enhance the predictive power of the EOB model for eccentric binaries and guide future SF calculations to determine higher-order eccentric corrections and additional EOB potentials such as .

Abstract

We analytically compute, through the six-and-a-half post-Newtonian order, the second-order-in-eccentricity piece of the Detweiler-Barack-Sago gauge-invariant redshift function for a small mass in eccentric orbit around a Schwarzschild black hole. Using the first law of mechanics for eccentric orbits [A. Le Tiec, Phys. Rev. D {\bf 92}, 084021 (2015)] we transcribe our result into a correspondingly accurate knowledge of the second radial potential of the effective-one-body formalism [A. Buonanno and T. Damour, Phys. Rev. D {\bf 59}, 084006 (1999)]. We compare our newly acquired analytical information to several different numerical self-force data and find good agreement, within estimated error bars. We also obtain, for the first time, independent analytical checks of the recently derived, comparable-mass fourth-post-Newtonian order dynamics [T. Damour, P. Jaranowski and G. Shaefer, Phys. Rev. D {\bf 89}, 064058 (2014)].

Paper Structure

This paper contains 9 sections, 55 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. High PN-order analytical computation of the self-force correction to the averaged redshift function along eccentric orbits
  3. Confirmation of recently derived 4PN results
  4. Confirmation of recently obtained 5PN and 5.5PN results
  5. Comparison with SF results on small-eccentricity orbits: $O(e^2)$-level
  6. Going beyond the $O(e^2)$-level
  7. Comparison with $O(e^4)$ information extracted from SF results
  8. Comparison with SF results on eccentric orbits
  9. Conclusions

Figures (3)

  • Figure 1: Panel (a): the successive PN-approximants to the function $\bar{d}(u_p)$ are compared with the SF numerical data of Ref. Akcay:2012ea. Panel (b): the numerical data of Akcay:2012ea are confronted with two different analytical fits, 1) PN-like one given in Eqs. (9.40) and (9.41) of Ref. Damour:2015isa and 2) the Padé-like fit of Eq. \ref{['bard_fit']}, on the interval $0.1\le u_p \le 1/6$.
  • Figure 2:
  • Figure 3: Numerical SF data points (from Ref.Akcay:2015pza) for $\delta U(p,e)$ are compared with the sum of $\delta U(p,0)$ (given by model 14 in Akcay:2012ea) and of the PN-expanded analytical prediction of Eqs. \ref{['DeltaU_e2']}, \ref{['DeltaU_e4']}, \ref{['DeltaU_e6']}. We consider the two extreme eccentricities listed in Table II of Akcay:2015pza, namely $e=0.05$ and $e=0.4$.