General relativistic dynamical tides in binary inspirals, without modes

A neutron star in an inspiraling binary system is tidally deformed by its companion, and the effect leaves a measurable imprint on the emitted gravitational waves. While the tidal interaction falls within the regime of static tides during the early stages of inspiral, a regime of dynamical tides takes over in the later stages. The description of dynamical tides found in the literature makes integral use of a spectral representation of the tidal deformation, in which it is expressed as a sum over the star's normal modes of vibration. This description is deeply rooted in Newtonian fluid mechanics and gravitation, and we point out that considerable obstacles manifest themselves in an extension to general relativity. To remedy this we propose an alternative, mode-less description of dynamical tides that can be formulated in both Newtonian and relativistic mechanics. Our description is based on a time-derivative expansion of the tidal dynamics. The tidal deformation is characterized by two sets of Love numbers: the static Love numbers $k_\ell$ and the dynamic Love numbers $\ddot{k}_\ell$. These are computed here for polytropic stellar models in both Newtonian gravity and general relativity. The time-derivative expansion of the tidal dynamics seems to preclude any attempt to capture an approach to resonance, which occurs when the frequency of the tidal field becomes equal to a normal-mode frequency. To overcome this limitation we propose a pragmatic extension of the time-derivative expansion which does capture an approach to resonance. We demonstrate that with this extension, our formulation of dynamical tides should be just as accurate as the $f$-mode truncation of the mode representation, in which the sum over modes is truncated to a single term involving the star's fundamental mode of vibration.