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

High-order terms in the post-Newtonian (PN) expansions of various quantities for compact binaries exhibit a combinatorial increase in complexity, with an ever-increasing number of terms, including more transcendentals and logarithms of the velocity, higher powers of these transcendentals and logarithms, and larger and larger rational numbers as coefficients. 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 [O(v 44 ), where v is the particle's orbital velocity] 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 wave propagation on the Schwarzschild spacetime (e.g., tail effects). For the modes with azimuthal number ℓ ≥ 7, 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 (and also in the size of the entire expression for the mode). The reduction in complexity becomes less dramatic for smaller ℓ, due to the structure of the expansion, and one only obtains purely rational coefficients up to some order, 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 in [Isoyama et al., Phys. Rev. D 87, 024010 (2013)] 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.

Eccentric-orbit extreme-mass-ratio-inspiral radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders

“…However, there does exist an alternate way forward, which allows for an easier calculation of complicated high-PN logarithms like L 9L2 (e) to high (finite) order in e 2 . In a private communication, Nathan Johnson-McDaniel revealed a means by which his circular-orbit S lm tail factorization [60] (based on earlier work in [61,65]) can be extended to an S lmn tail factorization for eccentric orbits. This lmn factorization can be combined with fitting methods to greatly simplify (relative to fitting alone) the process of computing certain logarithmic PN terms to arbitrary order in e 2 .…”

“…In BHPT, circular orbits correspond to n = 0, while for the quadrupole moment the circular orbit flux is determined by n = 2. Using Johnson-McDaniel's S lm tail factorization [60], it is possible to use BHPT to extract the circular-orbit limit of the entire leading-logarithm series. Indeed, we can infer from the discussion in Section IV of [29] that this limit is generated entirely by the quadrupole factor |S 22 | 2 , which can be written as…”

“…2. The perturbation theory eulerlog function for lmn modes (see [50,60,65]): for integers (k, j), which emerge with various values of j during the orbital average of log k (r) terms in the flux. We reuse the integer k here to match the index on T k , as we expect the relevant integrals (for integer leading/subleading logs) to appear at (3k)PN order.…”

Spin–orbit precession for eccentric black hole binaries at first order in the mass ratio

“…From one perspective, the compact body m 1 is accelerated away from a geodesic of the background spacetime of m 2 by a GSF, which may be split into a dissipative ("radiation reaction") and conservative piece with respect to time-reversal. The dissipative self-force at leading order in q has been known, in effect, since the 1970s, as it may be deduced from Teukolsky fluxes [22,[45][46][47]. By contrast, the more subtle consequences of the conservative self-force have only been explored in the last decade.…”

High precision gravitational self-force calculations and post-Newtonian implications

“…), all with rational coefficients composed of integers of finite extent. It is that finiteness which allows us to extract analytically exact coefficients from results of finite accuracy [22].…”