Dynamical tides during the inspiral of rapidly spinning neutron stars: Solutions beyond mode resonance

We investigate the dynamical tide in a gravitational wave (GW)-driven coalescing binary involving a neutron star (NS). The NS is assumed to spin rapidly, with its spin axis anti-aligned with the orbit. Such an NS may exist if the binary forms dynamically in a dense environment, and it can lead to a strong tide because the f-mode can be resonantly excited during the inspiral. We present a new analytical solution for the f-mode resonance by decomposing the tide into a resummed equilibrium component and a dynamical component that is excited only around resonance. This solution simplifies numerical implementations by avoiding the subtraction of two diverging terms. It also extends the solution's validity to frequencies beyond mode resonance. When the dynamical tide back reacts on the orbit, the commonly adopted effective Love number is insufficient because it does not capture the tidal torque on the orbit that dominates the back reaction during mode resonance. An additional dressing factor originating from the imaginary part of the Love number is introduced to model the torque. The dissipative interaction between the NS and the orbital mass multipoles is computed including the dynamical tide. Orbital phase shifts caused by the $l=3$ and $l=2$ f-modes can reach 0.5 and 10 radians at their respective resonances if the NS has a spin rate of 850 Hz. Because of the large impact of the dynamical tide, a linearized analytical description becomes insufficient. After mode excitation, the orbit cannot remain quasi-circular, and the eccentricity excited by the dynamical tide can approach $e\simeq 0.1$, leading to non-monotonic frequency evolution which breaks the stationary phase approximation commonly adopted by frequency-domain waveform constructions. The GW radiation from the excited f-mode alone can be detected with a signal-to-noise ratio exceeding unity with the next-generation detectors.