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

Abstract

(Abridged) High-order terms in the post-Newtonian (PN) expansions of various quantities for compact binaries exhibit a combinatorial increase in complexity, including ever-increasing numbers of transcendentals. Here we consider the gravitational wave energy flux at infinity from a point particle in a circular orbit around a Schwarzschild black hole, which is known to 22PN beyond the lowest-order Newtonian prediction, at which point each order has over 1000 terms. We introduce a factorization that considerably simplifies the spherical harmonic modes of the energy flux (and thus also the amplitudes of the spherical harmonic modes of the gravitational waves); it is likely that much of the complexity this factorization removes is due to curved-space wave propagation (e.g., tail effects). For the modes with azimuthal number l of 7 or greater, this factorization reduces the expressions for the modes that enter the 22PN total energy flux to pure integer PN series with rational coefficients, which amounts to a reduction of up to a factor of ~150 in the total number of terms in a given mode. The reduction in complexity becomes less dramatic for smaller l, due to the structure of the expansion, though the factorization is still able to remove all the half-integer PN terms. For the 22PN l = 2 modes, this factorization still reduces the total number of terms (and size) by a factor of ~10 and gives purely rational coefficients through 8PN. This factorization also improves the convergence of the series, though we find the exponential resummation introduced for the full energy flux by Isoyama et al. to be even more effective at improving the convergence of the individual modes, producing improvements of over four orders of magnitude over the original series for some modes. However, the exponential resummation is not as effective at simplifying the series, particularly for the higher-order modes.

18 pages, 4 figures, version accepted by PRD, including various minor corrections

References in corpus (6)

• Faithful Effective-One-Body waveforms of small-mass-ratio coalescing black-hole binaries
• Improved effective-one-body description of coalescing nonspinning black-hole binaries and its numerical-relativity completion
• Gravitational Waves from a Particle in Circular Orbits around a Schwarzschild Black Hole to the 22nd Post-Newtonian Order
• Two-body gravitational spin-orbit interaction at linear order in the mass ratio
• Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential
• Impact of the second order self-forces on the dephasing of the gravitational waves from quasi-circular extreme mass-ratio inspirals

Cited by in corpus (9)

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• 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
• Eccentric-orbit extreme-mass-ratio-inspiral radiation II: 1PN correction to leading-logarithm and subleading-logarithm flux sequences and the entire perturbative 4PN flux
• Tidal heating and torquing of the primary black hole in eccentric-orbit, non-spinning extreme-mass-ratio inspirals to 22PN order
• Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals to 9PN and $e^{16}$
• High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals
• High-order post-Newtonian expansion of the generalized redshift invariant for eccentric-orbit, equatorial extreme-mass-ratio inspirals with a spinning primary