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 at least one neutron star (NS). The deformed NS is assumed to spin rapidly, with its spin axis antialigned with the orbit. Such an NS may exist if the binary forms dynamically in a dense environment, and it can lead to a particularly strong tide because the NS 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 varying at the tidal forcing frequency and a dynamical component varying at the f-mode eigenfrequency that is excited only around mode resonance. This solution simplifies numerical implementations by avoiding the subtraction of two diverging terms as was done in previous analyses. It also extends the solution's validity to frequencies beyond mode resonance. When the dynamical tide back reacts on the orbit, we demonstrate that 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 therefore introduced to model the torque. The dissipative interaction between the NS and the orbital mass multipoles is computed including the dynamical tide and shown to be subdominant compared to the conservative energy transfer from the orbit to the NS modes. Our study shows that 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 and direction antialigned with the orbit. Because of the large impact of the $l=2$ dynamical tide, a linearized analytical description becomes insufficient, calling for future developments to incorporate higher-order corrections. After mode excitation, the orbit cannot remain quasicircular, and the eccentricity excited by the $l=2$ dynamical tide can approach nearly $e\simeq 0.1$, leading to nonmonotonic frequency evolution which breaks the stationary phase approximation commonly adopted by frequency domain phenomenological waveform constructions. Lastly, we demonstrate that the GW radiation from the resonantly excited f-mode alone can be detected with a signal-to-noise ratio exceeding unity at a distance of 50 Mpc with the next-generation GW detectors.