Numerically generated quasi-equilibrium orbits of black holes: Circular or eccentric?

TL;DR

This paper compares numerical quasi-equilibrium black hole binary configurations with third-order post-Newtonian approximations, finding that allowing slight eccentricity improves the fit between methods at larger orbital separations.

Contribution

It demonstrates that including small eccentricity in post-Newtonian models better matches numerical data for black hole binaries than assuming purely circular orbits.

Findings

Eccentric orbits (e ≈ 0.03-0.05) fit numerical data better than circular models.

Post-Newtonian approximations with eccentricity outperform effective one-body methods.

Numerical initial data may introduce spurious eccentricity into orbits.

Abstract

We make a comparison between results from numerically generated, quasi-equilibrium configurations of compact binary systems of black holes in close orbits, and results from the post-Newtonian approximation. The post-Newtonian results are accurate through third PN order (O(v/c)^6 beyond Newtonian gravity), and include rotational and spin-orbit effects, but are generalized to permit orbits of non-zero eccentricity. Both treatments ignore gravitational radiation reaction. The energy E and angular momentum J of a given configuration are compared between the two methods as a function of the orbital angular frequency \Omega. For small \Omega, corresponding to orbital separations a factor of two larger than that of the innermost stable orbit, we find that, if the orbit is permitted to be slightly eccentric, with e ranging from \approx 0.03 to \approx 0.05, and with the two objects initially located at the orbital apocenter (maximum separation), our PN formulae give much better fits to the numerically generated data than do any circular-orbit PN methods, including various ``effective one-body'' resummation techniques. We speculate that the approximations made in solving the initial value equations of general relativity numerically may introduce a spurious eccentricity into the orbits.