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New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole

Donato Bini, Thibault Damour, Andrea Geralico

TL;DR

The paper advances analytic control of the conservative self-force for eccentric orbits around a Schwarzschild black hole by deriving high-order eccentricity expansions of the Detweiler-Barack-Sago redshift invariant in the $1$SF framework. It combines Regge-Wheeler-Zerilli perturbation theory with MST techniques and the eccentric first law to obtain $\delta U^{e^2}$ and $\delta U^{e^4}$ up to $9.5$PN, and extends to $e^{20}$ at $4$PN, translating these results into the EOB potentials $\bar{d}(u)$, $\rho(u)$, and $q(u)$. The study provides explicit analytic expressions and PN-remainder estimates, and validates them against state-of-the-art SF numerical data, finding overall good agreement while highlighting convergence limitations for $q(u)$. It also delivers $4$PN results for higher-$e$ powers through $e^{20}$ and computes related $Q$-potential coefficients, strengthening the PN-SF-EOB bridge and informing future hybrid analytic-numeric approaches and domain explorations beyond the ISCO. The findings have practical implications for precise gravitational-wave modeling of eccentric binaries and for guiding further SF studies using hyperbolic scattering to probe strong-field regions.

Abstract

We raise the analytical knowledge of the eccentricity-expansion of the Detweiler-Barack-Sago redshift invariant in a Schwarzschild spacetime up to the 9.5th post-Newtonian order (included) for the $e^2$ and $e^4$ contributions, and up to the 4th post-Newtonian order for the higher eccentricity contributions through $e^{20}$. We convert this information into an analytical knowledge of the effective-one-body radial potentials $\bar d(u)$, $ρ(u)$ and $q(u)$ through the 9.5th post-Newtonian order. We find that our analytical results are compatible with current corresponding numerical self-force data.

New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole

TL;DR

The paper advances analytic control of the conservative self-force for eccentric orbits around a Schwarzschild black hole by deriving high-order eccentricity expansions of the Detweiler-Barack-Sago redshift invariant in the SF framework. It combines Regge-Wheeler-Zerilli perturbation theory with MST techniques and the eccentric first law to obtain and up to PN, and extends to at PN, translating these results into the EOB potentials , , and . The study provides explicit analytic expressions and PN-remainder estimates, and validates them against state-of-the-art SF numerical data, finding overall good agreement while highlighting convergence limitations for . It also delivers PN results for higher- powers through and computes related -potential coefficients, strengthening the PN-SF-EOB bridge and informing future hybrid analytic-numeric approaches and domain explorations beyond the ISCO. The findings have practical implications for precise gravitational-wave modeling of eccentric binaries and for guiding further SF studies using hyperbolic scattering to probe strong-field regions.

Abstract

We raise the analytical knowledge of the eccentricity-expansion of the Detweiler-Barack-Sago redshift invariant in a Schwarzschild spacetime up to the 9.5th post-Newtonian order (included) for the and contributions, and up to the 4th post-Newtonian order for the higher eccentricity contributions through . We convert this information into an analytical knowledge of the effective-one-body radial potentials , and through the 9.5th post-Newtonian order. We find that our analytical results are compatible with current corresponding numerical self-force data.

Paper Structure

This paper contains 12 sections, 79 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Novel analytical results for $\delta U^{e^2}$ and $\delta U^{e^4}$ up to the 9.5PN order
  3. Estimating the order of magnitude of the coefficients of PN expansions
  4. Comparing 9.5PN-accurate theoretical results to self-force numerical data
  5. Analytical $4$PN results for $\delta U^{e^n}$ up to $n=20$
  6. Discussion
  7. Combined use of RWZ approach, PN solutions and MST technique
  8. Unperturbed particle motion
  9. Source terms
  10. Green function
  11. Computing $h_{uk}$
  12. Coefficients of the PN expansions of the functions $a(u_3)$, $\bar{d}(u_3)$, $\rho(u_3)$, and $q(u_3)$ (where $u_3 \equiv 3 u$).

Figures (2)

  • Figure 1: The behavior of various PN-approximants to the EOB functions $\bar{d}$ and $q$ is shown in panels (a) and (b), respectively, and compared to a sample of numerical data points from Akcay:2015pjz.
  • Figure 2: The (base-10) logarithms of twice the fractional theoretical PN errors on $\bar{d}$ (panel a), and $q$ (panel b), for $N=[3,4,6.5,9.5,14.5]$ (see text for details).