Gravitational radiation from inspiralling compact objects: Spin-spin effects completed at the next-to-leading post-Newtonian order

Using the gravitational potential and source multipole moments bilinear in the spins, first computed to next-to- leading order (NLO) in the post-Newtonian expansion within the effective field theory (EFT) framework, we complete here the derivation of the dynamical invariants and flux-balance equations, including energy and angular momentum. We use these results to calculate spin-spin effects in the orbital frequency and accumulated phase to NLO for circular orbits. We also derive the linear momentum and center-of-mass fluxes and associated kick velocity, to the highest relevant post- Newtonian order. We explicitly demonstrate the equivalence between the quadratic-in-spin source multipoles obtained using the EFT formalism and those rederived later with more traditional tools, leading to perfect agreement for spin-spin radiative observables to NLO among both approaches.