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

TL;DR

This paper introduces a factorization method that simplifies the high-order post-Newtonian expansions of gravitational wave energy flux modes for a particle orbiting a Schwarzschild black hole, significantly reducing complexity and improving convergence.

Contribution

The authors present a novel factorization technique that simplifies the spherical harmonic modes of the energy flux, reducing the number of terms and removing half-integer PN terms, especially at high PN orders.

Findings

Reduces the number of terms in high-l modes by up to a factor of 150.

Removes all half-integer PN terms from the modes.

Improves convergence of the series, surpassing exponential resummation in some cases.

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.