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We investigate the dynamics of spinning binaries of compact objects at thenext-to-leading order in the quadratic-in-spin effects, which corresponds tothe third post-Newtonian order (3PN). Using a Dixon-type multipolar formalismfor spinning point particles endowed with spin-induced quadrupoles andcomputing iteratively in harmonic coordinates the relevant pieces of the PNmetric within the near zone, we derive the post-Newtonian equations of motionas well as the equations of spin precession. We find full equivalence withavailable results. We then focus on the far-zone field produced by thosesystems and obtain the previously unknown 3PN spin contributions to thegravitational-wave energy flux by means of the multipolar post-Minkowskian(MPM) wave generation formalism. Our results are presented in thecenter-of-mass frame for generic orbits, before being further specialized tothe case of spin-aligned, circular orbits. We derive the orbital phase of thebinary based on the energy balance equation and briefly discuss the relevanceof the new terms.

Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3PN order