Nonlinear mode-tide coupling in coalescing binary neutron stars with relativistic corrections

Abstract

We compute the internal modes of a nonspinning neutron star and its tidal metric perturbation in general relativity, and determine the effect of relativistic corrections to the modes on mode coupling. Claims have been made that a new hydrodynamic instability can occur in a neutron star in a binary neutron star system triggered by the nonlinear coupling of the companion's tidal field to pairs of p-modes and g-modes in it as the binary inspirals toward merger. This "PG" instability may be significant since it can influence the binary's inspiral phase by extracting orbital energy, thereby potentially causing large deviations in their gravitational waveforms from those predicted by theoretical models that do not account for it. This can result in incorrect parameter estimation, at best, or mergers going undetected, at worst, owing to the use of deficient waveform models. On the other hand, better modeling of this instability and its effect on binary orbits can unravel a new phenomenon and shed light on stellar instabilities, via gravitational wave observations. So far, all mode-tide coupling instability studies have been formulated in Newtonian perturbation theory. Neutron stars are compact objects, so relativistic corrections might be important. We present and test a new code to calculate the relativistic eigenmodes of nonrotating relativistic stars. We use these relativistic tide and neutron star eigenmodes to compute the mode-tide coupling strength (MTCS) for a few selected equations of state. The MTCS thus calculated can be at most tens of percent different from its purely Newtonian value, but we confirm the dependencies on orbital separation and equation of state found by Newtonian calculations. For some equations of state, the MTCS is very sensitive to the neutron star crust region, demonstrating the importance of treating this region accurately.