High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals

Abstract

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $⟨u^t{⟩}_{\tau }$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory (BHPT) using Regge-Wheeler-Zerilli (RWZ) gauge. Our \textsc{Mathematica} code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi (MST) analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ equation for arbitrarily high $l$. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and $e^{20}$. Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.

26 pages, 2 figures

References in corpus (17)

• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• FastEMRIWaveforms: New tools for millihertz gravitational-wave data analysis
• Gravitational self-force correction to the innermost stable circular orbit of a Schwarzschild black hole
• Fourth post-Newtonian effective one-body dynamics
• High-Order Post-Newtonian Fit of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry
• Sixth post-Newtonian local-in-time dynamics of binary systems
• Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism
• 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
• Two-body gravitational spin-orbit interaction at linear order in the mass ratio
• Tidal invariants for compact binaries on quasi-circular orbits
• Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild 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
• 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)

• Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals to 9PN and $e^{16}$