Eccentric-orbit extreme-mass-ratio inspirals: Analytic expansion in the conservative sector of black hole perturbation theory to high PN order

We present new advances in determining analytic PN series for extreme-mass-ratio inspirals on a Schwarzschild background. We focus on a pair of gauge invariant quantities, the redshift invariant and spin-precession invariant, which help encode the conservative dynamics of these binaries under first-order black hole perturbation theory. The PN series are pursued using direct expansion of the MST solutions to the RWZ equations, along with a PN ansatz solution for RWZ equations with large mode number $l$. We compute the redshift invariant to 8.5PN through 20th order in eccentricity, finding that at multiple PN orders the eccentricity expansion exhibits a structured appearance which permits the extraction of a compact eccentricity function. Then, we utilize similar techniques to determine the spin-precession invariant to 6.5PN through 16th order in eccentricity. We conclude by discussing future applications to the more complicated Kerr problem.