Finding high-order analytic post-Newtonian parameters from a high-precision numerical self-force calculation

Abstract

We present a novel analytic extraction of high-order post-Newtonian (pN) parameters that govern quasi-circular binary systems. Coefficients in the pN expansion of the energy of a binary system can be found from corresponding coefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change ∆U in the redshift factor of a circular orbit at fixed angular velocity. Remarkably, by computing this essentially gauge-invariant quantity to accuracy greater than one part in 10 225 , and by assuming that a subset of pN coefficients are rational numbers or products of π and a rational, we obtain the exact analytic coefficients. We find the previously unexpected result that the postNewtonian expansion of ∆U (and of the change ∆Ω in the angular velocity at fixed redshift factor) have conservative terms at half-integral pN order beginning with a 5.5 pN term. This implies the existence of a corresponding 5.5 pN term in the expansion of the energy of a binary system.Coefficients in the pN series that do not belong to the subset just described are obtained to accuracy better than 1 part in 10 265−23n at nth pN order. We work in a radiation gauge, finding the radiative part of the metric perturbation from the gauge-invariant Weyl scalar ψ0 via a Hertz potential. We use mode-sum renormalization, and find high-order renormalization coefficients by matching a series in L = ℓ+1/2 to the large-L behavior of the expression for ∆U . The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Schwarzschild gauge. * a.g

Experimental mathematics meets gravitational self-force

“…We refer the reader to [21,[51][52][53][57][58][59] for details pertaining to the renormalization procedure.…”

“…Here we give the basics of the method we use to calculate ∆U (and its precise definition)-see [21,[51][52][53] for further details. We calculate ∆U in a modified radiation gauge, where ℓ ≥ 2 modes are calculated in an outgoing radiation gauge (with h αβ n α = 0 and h = 0, where n α is the ingoing null vector and h αβ and h are the metric perturbation and its trace, respectively) and the lower ones (ℓ = 0, 1) are calculated in the asymptotically flat Schwarzschild gauge.…”

“…It has recently been shown (see [21,34,37]) how the overlap region between the self-force formalism and the PN approximation can be explored using very high accuracy numerical results, which make it relatively easy to extract high-order PN coefficients that are currently out of reach of standard PN calculations. The coefficients obtained using this numerical extraction method have then been checked by independent analytical calculations.…”

“…Shah, Friedman, and Whiting (SFW) [21] worked solely on the expansion of the full ∆U and obtained analytic terms for the simplest coefficients, which are purely rational, or a rational times π, where they could easily identify the analytic form from a large enough number of digits. They also present three additional analytic expressions for more complicated higherorder terms (in the note added), which were obtained by the first author of this paper using an integer relation algorithm.…”

“…In [19], Bini and Damour calculated the nonlogarithmic portion 4PN coefficient analytically by using analytical solutions of the Regge-Wheeler-Zerilli equations; the logarithmic term had already been computed by Damour in [20]. Shah, Friedman, and Whiting (SFW) [21] calculated higher order PN coefficients, up to 10.5PN order, by calculating high-precision numerical values in a modified radiation gauge at very large radii and fitting them to a PN series to extract the coefficients. They found that half-integer terms started at 5.5PN, which they verified analytically (and was also verified using standard PN methods in [22,23]).…”