Eccentric binary black holes with spin via the direct integration of the post-Newtonian equations of motion

We integrate the third and a half post-Newtonian equations of motion for a fully generic binary black hole system, allowing both for noncircular orbits, and for one or both of the black holes to spin, in any orientation. Using the second post-Newtonian order expression beyond the leading order quadrupole formula, we study the gravitational waveforms produced from such systems. Our results are validated by comparing to Taylor T4 in the aligned-spin circular cases, and the additional effects and modulations introduced by the eccentricity and the spins are analyzed. We use the framework to evaluate the evolution of eccentricity, and trace its contributions to source terms corresponding to the different definitions. Finally, we discuss how this direct integration equations-of-motion code may be relevant to existing and upcoming gravitational wave detectors, showing fully generic, precessing, eccentric gravitational waveforms from a fiducial binary system with the orbital plane and spin precession, and the eccentricity reduction.