Comparison between self-force and post-Newtonian dynamics: Beyond circular orbits

Abstract

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modeling the dynamics of compact binary systems. Comparison of their results in an overlapping domain of validity provides a crucial test for both methods and can be used to enhance their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first time, we extend such comparisons to noncircular orbits-specifically, to a system of two nonspinning objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged quantity hUi that generalizes Detweiler's redshift invariant. The functional relationship hUiðΩ r ; Ω ϕ Þ, where Ω r and Ω ϕ are the frequencies of the radial and azimuthal motions, is an invariant characteristic of the conservative dynamics. We compute hUiðΩ r ; Ω ϕ Þ numerically through linear order in the mass ratio q, using a GSF code which is based on a frequency-domain treatment of the linearized Einstein equations in the Lorenz gauge. We also derive hUiðΩ r ; Ω ϕ Þ analytically through 3PN order, for an arbitrary q, using the known near-zone 3PN metric and the generalized quasi-Keplerian representation of the motion. We demonstrate that the OðqÞ piece of the analytical PN prediction is perfectly consistent with the numerical GSF results, and we use the latter to estimate yet unknown pieces of the 4PN expression at OðqÞ.

Numerical computation of the effective-one-body potentialqusing self-force results

“…The details of the computation for U (1) (v, e) are thoroughly explained in Refs. [12], [45] so here, we focus on the fitting procedure. We used the following four polynomials in e 2 for our fits:…”

“…Indeed, attempting to do so using the procedure described in Refs. [12] and [13], i.e. by fitting v 7 , v 7 log v, v 7 log 2 v and higher order terms to the residual, yields inconclusive values for the coefficients of the fitting functions, which are the unknown higher-order pN parameters.…”

“…(1) (v) by fitting polynomials in powers of e 2 to the numerical data for U (1) (v, e) at each v. We use two independent approaches: (i) fitting polynomials directly to U (1) (v, e) data obtained from the C-based Lorenz-gauge code of Refs. [12,45], (ii) using the Mathematica-based Teukolsky-MST-CCK code of Ref. [13] which extracts the e 0 , e 2 and e 4 dependence of the numerically computed multipole l modes of h R uu then constructs power-law fits to the resulting three separate sets of mode data.…”

“…The GSF correction to the redshift for eccentric orbits was first calculated by Barack and Sago [25]. Much-improved results have recently been produced by the authors with frequency-domain methods using both a Lorenz-gauge approach [12], and a radiation-gauge approach where the metric perturbation is reconstructed from the Weyl scalars [13]. Combining these methods, we determine the EOBd and q potentials for dimensionless binary separations of 6 r ≤ 1200 (see Sec.…”

“…The aim of this paper is to utilize recent technological advances in eccentric-orbit self-force computations [12,13] to determine the linear-in-mass-ratio contributions to the potentials in the EOB Hamiltonian for moderately eccentric, non-spinning binaries. Currently, the numerical relativity calibrated EOB-based wave templates (EOBNR [7][8][9]) -in use in the detection pipeline of the Advanced LIGO detector -only model quasicircular inspirals.…”