The stability of tidally deformed neutron stars to three- and four-mode coupling

TL;DR

This paper demonstrates that the previously suggested tidal instability in neutron stars due to three-mode coupling is canceled out by four-mode interactions, confirming the stability of neutron stars during inspiral and its negligible effect on gravitational wave signals.

Contribution

The study introduces a volume-preserving coordinate transformation to relate four-mode couplings to three-mode couplings, revealing a near-exact cancellation that stabilizes neutron stars against tidal mode instabilities.

Findings

Four-mode interactions cancel three-mode destabilizing effects

The equilibrium tide remains stable against decay into daughter modes

Instability timescales are longer than gravitational-wave inspiral times

Abstract

It has recently been suggested that the tidal deformation of a neutron star excites daughter p- and g-modes to large amplitudes via a quasi-static instability. This would remove energy from the tidal bulge, resulting in dissipation and possibly affecting the phase evolution of inspiralling binary neutron stars and hence the extraction of binary parameters from gravitational wave observations. This instability appears to arise because of a large three-mode interaction among the tidal mode and high-order p- and g-modes of similar radial wavenumber. We show that additional four-mode interactions enter into the analysis at the same order as the three-mode terms previously considered. We compute these four-mode couplings by finding a volume-preserving coordinate transformation that relates the energy of a tidally deformed star to that of a radially perturbed spherical star. Using this method, we relate the four-mode coupling to three-mode couplings and show that there is a near-exact cancellation between the destabilizing effect of the three-mode interactions and the stabilizing effect of the four-mode interaction. We then show that the equilibrium tide is stable against the quasi-static decay into daughter p- and g-modes to leading order. The leading deviation from the quasi-static approximation due to orbital motion of the binary is considered; while it may slightly spoil the near-cancellation, any resulting instability timescale is at least of order the gravitational-wave inspiral time. We conclude that the p-/g-mode coupling does not lead to a quasi-static instability, and does not impact the phase evolution of gravitational waves from binary neutron stars.