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

Abstract

We derive new terms in the post-Newtonian (PN) expansion of the generalized redshift invariant $⟨u^t{⟩}_{\tau }$ 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.

23 pages, 2 figures

References in corpus (17)

• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• Fourth post-Newtonian effective one-body dynamics
• Sixth post-Newtonian local-in-time dynamics of binary systems
• Inspiralling compact binaries in quasi-elliptical orbits: The complete third post-Newtonian energy flux
• Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism
• Gravitational Waves from a Particle in Circular Orbits around a Schwarzschild Black Hole to the 22nd Post-Newtonian Order
• Gravitational self-force corrections to two-body tidal interactions and the effective one-body formalism
• Tail effects in the third post-Newtonian gravitational wave energy flux of compact binaries in quasi-elliptical orbits
• Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole to the 11th Post-Newtonian Order
• Completion of metric reconstruction for a particle orbiting a Kerr black hole
• Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits
• Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio
• Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription
• Accuracy of the post-Newtonian approximation. II. Optimal asymptotic expansion of the energy flux for quasicircular, extreme mass-ratio inspirals into a Kerr black hole
• Gyroscope precession along bound equatorial plane orbits around a Kerr black hole
• Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting
• Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

Cited by in corpus (1)

• Constants of motion in gravitational self-force theory