Impact of nonlinearities on relativistic dynamical tides in compact binary inspirals

The tidal deformation of a neutron star in a binary inspiral driven by the emission of gravitational waves affects the orbital dynamics and produces a measurable modulation of the waves. Late in the inspiral, a regime of dynamical tides takes over from a prior regime of static tides. A recent analysis by Yu et al. [M.N.R.A.S. 519, 4325 (2022)] reveals that nonlinear aspects of the tidal interaction are important during the regime of dynamical tides. Their theoretical framework is grounded in Newtonian gravity and fluid mechanics, and relies on a representation of the tidal deformation in terms of the star's normal modes of vibration. We confirm their observation in a general relativistic treatment of the tidal deformation of a neutron star, without relying on a mode representation of this deformation. The starting point of our description is a simultaneous time-derivative and nonlinear expansion of the tidal deformation, expressed in terms of three encapsulating constants, the static $k_2$, dynamic $\ddot{k}_2$, and nonlinear $p_2$ tidal constants. We describe the neutron star's deformation in terms of a well-defined quadrupole moment tensor, which is related to the tidal quadrupole moment through a frequency-domain response function $\tilde{k}_2(\omega)$. In a pragmatic extension of our simultaneous expansion, we express this in a form proportional to $(1-\omega^2/\omega_*^2)^{-1}$, the characteristic response of a harmonic oscillator subjected to a driving force of frequency $\omega$, with a natural-frequency parameter $\omega_*$ constructed from the tidal constants. We compute these for polytropic stellar models, and show that the nonlinear constant $p_2$ lowers the frequency parameter by as much as 15% relative to an estimation based on a purely linear treatment of the tidal deformation.