High-order post-Newtonian expansion of the generalized redshift invariant for eccentric-orbit, equatorial extreme-mass-ratio inspirals with a spinning primary

TL;DR

This paper analytically derives high-order post-Newtonian terms for the redshift invariant in eccentric, equatorial extreme-mass-ratio inspirals around spinning black holes, enhancing gravitational wave modeling accuracy.

Contribution

It introduces new high-order eccentricity and spin-dependent terms in the PN expansion of the redshift invariant using advanced perturbation techniques.

Findings

PN series expanded to $e^{16}$ at 6PN order

Many terms simplified to closed-form functions of eccentricity

Numerical analysis shows convergence improvements over Schwarzschild case

Abstract

We derive new terms in the post-Newtonian (PN) expansion of the generalized redshift invariant $\langle u^t \rangle_\tau$ for a small body in eccentric, equatorial orbit about a massive Kerr black hole. The series is computed analytically using the Teukolsky formalism for first-order black hole perturbation theory (BHPT), along with the CCK method for metric reconstruction using the Hertz potential in ingoing radiation gauge. Modal contributions with small values of $l$ are derived via the semi-analytic solution of Mano-Suzuki-Takasugi (MST), while the remaining values of $l$ to infinity are determined via direct expansion of the Teukolsky equation. Each PN order is calculated as a series in eccentricity $e$ but kept exact in the primary black hole's spin parameter $a$. In total, the PN terms are expanded to $e^{16}$ through 6PN relative order, and separately to $e^{10}$ through 8PN relative order. Upon grouping eccentricity coefficients by spin dependence, we find that many resulting component terms can be simplified to closed-form functions of eccentricity, in close analogy to corresponding terms derived previously in the Schwarzschild limit. We use numerical calculations to compare convergence of the full series to its Schwarzschild counterpart and discuss implications for gravitational wave analysis.