Analytic self-force calculations in the post-Newtonian regime: eccentric orbits on a Schwarzschild background

TL;DR

This work develops an analytic PN framework for gravitational self-force (GSF) calculations sourced by eccentric orbits around a Schwarzschild black hole, operating in Regge-Wheeler-Zerilli (RWZ) gauge. It combines a frequency-domain extended homogeneous-solution approach with a PN/small-eccentricity expansion to obtain closed-form retarded metric perturbations for all $\ell\ge 2$ through 4PN and $e^{10}$, including large-$\ell$ expressions and a re-summation that captures $e\to 1$ behavior. The authors compute the gauge-invariant generalized redshift $\langle U \rangle_{gsf}$ and connect their results to the EOB potential $\hat Q$, validating against recent PN and numerical self-force data and achieving strong agreement across multiple benchmarks. The methodology yields high-precision analytic benchmarks for calibrating waveform models and provides a pathway toward Kerr extensions and broader invariant calculations in the PN/GSF overlap regime.

Abstract

We present a method for solving the first-order field equations in a post-Newtonian (PN) expansion. Our calculations generalize work of Bini and Damour and subsequently Kavanagh et al., to consider eccentric orbits on a Schwarzschild background. We derive expressions for the retarded metric perturbation at the location of the particle for all $\ell$-modes. We find that, despite first appearances, the Regge-Wheeler gauge metric perturbation is $C^0$ at the particle for all $\ell$. As a first use of our solutions, we compute the gauge-invariant quantity $\langle U \rangle$ through 4PN while simultaneously expanding in eccentricity through $e^{10}$. By anticipating the $e\to 1$ singular behavior at each PN order, we greatly improve the accuracy of our results for large $e$. We use $\langle U \rangle$ to find 4PN contributions to the effective one body potential $\hat Q$ through $e^{10}$ and at linear order in the mass-ratio.

Figures (2)

• Figure 1: The effect of re-summing the small-$e$ expansion at each PN order as seen by comparing to numerical data, all computed at $p=1000$. We see that especially as eccentricity increases, our re-summation greatly improves convergence. Also, note the consistency of our convergence throughout the orbit. Our results are no less effective at periapsis than apoapsis. The dips in the residuals are from zero crossings, and not meaningful. See the discussion in the text for more details.
• Figure 2: Comparison of our PN expressions with numerical data from Table II of Akcay et al. AkcaETC15. Dots are numerical values and lines are our analytic calculations. The top row is $\log_{10}$ of the absolute value of the full $\langle U \rangle_{\rm gsf}$. Successive rows show residuals after subtracting each term in the PN series. The consistency of our agreement out to $e=0.4$ is only possible because of our re-summation of the small-$e$ expansion. Unevenness of the final residuals for $p = 100$ is likely a numerical artifact.