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

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 $eto 1$ 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.