Lorenz gauge gravitational self-force calculations of eccentric binaries using a frequency domain procedure

Abstract

We present an algorithm for calculating the metric perturbations and gravitational self-force for extreme-mass-ratio inspirals (EMRIs) with eccentric orbits. The massive black hole is taken to be Schwarzschild and metric perturbations are computed in Lorenz gauge. The perturbation equations are solved as coupled systems of ordinary differential equations in the frequency domain. Accurate local behavior of the metric is attained through use of the method of extended homogeneous solutions and mode-sum regularization is used to find the self-force. We focus on calculating the self-force with sufficient accuracy to ensure its error contributions to the phase in a long term orbital evolution will be $\delta\Phi \lesssim 10^{-2}$ radians. This requires the orbit-averaged force to have fractional errors $\lesssim 10^{-8}$ and the oscillatory part of the self-force to have errors $\lesssim 10^{-3}$ (a level frequently easily exceeded). Our code meets this error requirement in the oscillatory part, extending the reach to EMRIs with eccentricities of $e \lesssim 0.8$, if augmented by use of fluxes for the orbit-averaged force, or to eccentricities of $e \lesssim 0.5$ when used as a stand-alone code. Further, we demonstrate accurate calculations up to orbital separations of $a \simeq 100 M$, beyond that required for EMRI models and useful for comparison with post-Newtonian theory. Our principal developments include (1) use of fully constrained field equations, (2) discovery of analytic solutions for even-parity static modes, (3) finding a pre-conditioning technique for outer homogeneous solutions, (4) adaptive use of quad-precision and (5) jump conditions to handle near-static modes, and (6) a hybrid scheme for high eccentricities

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

“…No published numerical data exist to allow comparison beyond p ¼ 20. (Reference [56] gives results for e ≤ 0.7 and p ≤ 90, but these are for the GSF components, not for hUi gsf . )…”

“…[56], and against yet unpublished redshift data calculated by van de Meent [57] (using a very different frequencydomain method based on a semianalytical treatment of Teukolsky's equation [58]). These comparisons strongly favor the frequency-domain data in the table.…”

“…The issue of nearly static modes has been addressed in a very recent paper by Osburn et al [56], who proposed additional mitigation methods. These may be used to improve the performance of weak-field calculations in future work.…”

“…Some improvement may be achieved using the method of Ref. [56], which is a slightly more advanced variant of our method. More significant improvements may have to await the development of eccentric-orbit GSF codes based on the Teukolsky equation [58,82].…”

Gravitational self-force on generic bound geodesics in Kerr spacetime

“…At the next order, the corrections to the equations of motion due to the gravitational field generated by the secondary can be collected into an effective force term perturbing the geodesic equation, the gravitational self-force (GSF). Since this force is small the evolutionary timescale (t insp = O(η −1 )) of an EMRI is much larger than the orbital timescale arXiv:1711.09607v1 [gr-qc] 27 Nov 2017 (t orb = O(1)). This hierarchy of timescales can be exploited to simplify the evolution of EMRIs by using a two timescale expansion.…”