Gravitational radiation reaction: Spin-spin effects in the inspiral of coalescing compact binaries

We study the equations of motion for binary systems of spinning compact bodies using post-Newtonian equations of motion for fluid bodies that include gravitational radiation reaction at 2.5 and 3.5 post-Newtonian (PN) order (O[(v/c)5] and O[(v/c) 7] beyond Newtonian order). In particular, we derive, from the first principles, the leading 3.5PN order radiation reaction coupled to the spin-spin effects on both the orbital motion and the evolution of spins. We find that the back radiation due to the emission of gravitational waves causes a 3.5 post-Newtonian order spin-spin induced precession of the individual spins. This is different from the spin-orbit coupling which has no effect on the individual spins at 3.5PN order. Then, from the equations of motion and evolution of individual spins, we derive the energy and angular momentum loss induced by the spin-spin effects. When comparing our result to the radiative flux derived in the radiation zone by Kidder et al., we find that the total energy and angular momentum loss from our calculation precisely balances the radiative flux of those quantities in the radiation zone.