Nonlinear hydrodynamics in spinning neutron stars: Theoretical universal relations and equilibrium solutions

We study tides during the inspiral of a binary neutron star (BNS) system, including nonlinear hydrodynamical interactions. Using an affine approximation that treats the perturbed NS as an ellipsoid, we analytically derive coupling coefficients among the f-modes and the radial mode to the four-wave order (i.e., next-to-next-to-leading order) in the Hamiltonian, allowing for arbitrary rotation of the background star. Our model reveals a series of universal relations from first-principles arguments. Besides the well-known relations, we show that the three-wave (next-to-leading-order) interaction coefficients are fully determined by the properties of the linear tide. Therefore, they do not probe new physics of the NS. Nonetheless, not including the three-wave nonlinear tides can lead to significant systematic errors in the gravitational waveform. We support this claim via a hybrid approach that simultaneously captures mode resonances expected in Newtonian hydrodynamics and is consistent with relativistic calculations in the low-frequency expansion. The nonlinear tide in a single NS can cause a phase shift of around 1.7 radians accumulated up to merger compared to the linear tide model; for a binary of similar masses, the phase shift is approximately doubled. Our calculation extends to four-wave interactions, which, for a slowly spinning NS, provide only small corrections and are subdominant compared to the tidal back-reaction on the orbit. For a rapidly rotating NS, the nonlinear centrifugal drive of the f-mode and the four-wave anharmonicity provides a window to study the adiabatic exponent related to internal buoyancy that cannot be probed by the linear and three-wave tides in slowly spinning systems. The anharmonicity cannot lead to resonance locking of the f-mode.