$f$-mode Imprints in Gravitational Waves from Coalescing Binaries involving Aligned Spinning Neutron Stars

The excitation of $f$-mode in a neutron star member of coalescing binaries accelerates the merger course, and thereby introduces a phase shift in the gravitational waveform. Emphasising on the tidal phase shift by aligned, rotating stars, we provide an accurate, yet economical, method to generate $f$-mode-involved, pre-merger waveforms using realistic spin-modulated $f$-mode frequencies for some viable equations of state. We find for slow-rotating stars that the dephasing effects of the dynamical tides can be uniquely, EOS-independently determined by the direct observables (chirp mass ${\cal M}$, symmetric ratio $\eta$ and the mutual tidal deformability ${\tilde \Lambda}$), while this universality is gradually lost for increasing spin. For binaries with fast rotating members ($\gtrsim800\text{ Hz}$) the phase shift due to $f$-mode will exceed the uncertainty in the waveform phase at reasonable signal-to-noise ($\rho=25$) and cutoff frequency of $\gtrsim400\text{ Hz}$. Assuming a high cutoff frequency of $10^3\text{ Hz}$ and fast ($\gtrsim800\text{ Hz}$) members, a significant phase shift of $\gtrsim100$ rads has been found. For systems involving a rapidly-spinning star (potentially the secondary of GW190814), neglecting $f$-mode effect in the waveform templates can therefore lead to considerable systemic errors in the relevant analysis. In particular, the dephasing due to $f$-mode is larger than that caused by equilibrium tides by a factor of $\sim5$, which may lead to a considerably overestimated tidal deformability if dynamical tidal contribution is not accounted. The possibility of accompanying precursors flares due to $f$-mode excitation is also discussed.