Nonlinear hydrodynamics in spinning neutron stars: Theoretical universal relations and equilibrium solutions
We study tides during the inspiral of a binary neutron star system, including nonlinear hydrodynamical interactions. Using an affine approximation that treats the perturbed neutron star (NS) as an ellipsoid, we analytically derive coupling coefficients among the quadrupolar f-modes and the radial mode to the four-wave order (next-to-next-to-leading order) in the Hamiltonian, allowing for arbitrary (aligned or anti-aligned) spin of the background star. Our model reveals a series of universal relations from first-principles arguments. We show that the three-wave (next-to-leading-order) interaction coefficients in a non-spinning star are fully determined by the properties of the linear tide. They do not probe new physics of the NS. Nonetheless, three-wave nonlinear tides are significant corrections to the gravitational waveform. We support this 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 1.8 radians accumulated up to merger compared to the linear tide model; for a binary, the phase shift is approximately doubled. In a low-frequency expansion, the nonlinear tide is degenerate with the finite-frequency correction of the linear tide, introducing systematic bias when ignored. Our calculation extends to four-wave interactions, which, for a slowly spinning neutron star, provide only small corrections. For a rapidly rotating neutron star, the nonlinear centrifugal drive of the f-mode provides a window to study the internal buoyancy that cannot be probed by the linear and three-wave f-mode tides in slowly spinning systems. The four-wave anharmonicity cannot lead to resonance locking of the f-mode.