Rotational-tidal phasing of the binary neutron star waveform

Tidal forces cause inspiralling binary neutron stars to deform, leaving a measurable imprint on the gravitational waves they emit. The induced stellar multipoles are an added source of gravitational radiation and modify the orbital dynamics, producing a slight acceleration of the coalescence which manifests as a phase shift in the waveform relative to point-particles. The dominant piece of this tidal phase comes from the mass quadrupoles, which contribute at fifth post-Newtonian order (5PN). Current quadrupoles and mass octupoles contribute at higher orders. For spinning neutron stars, additional multipole moments are induced by nonlinear couplings between spin and tides. We calculate these rotational-tidal deformations assuming the stars are rotating slowly and the tides are weak and quasi-stationary. The stellar multipole moments are read off from an asymptotically flat metric that encodes the difference between their tidal response and a black hole's. The multipoles are subsequently inserted into post-Newtonian formulas for the orbit and the gravitational radiation. We find that, at leading order, the rotational-tidal deformations make a 6.5PN contribution to the tidal phase. Their effect on the waveform is thus larger than that of the mass octupoles, and nearly as large as that of the current quadrupoles, in systems with non-negligible spin.