The bar-mode instability in differentially rotating neutron stars: Simulations in full general relativity

We study the dynamical stability against bar-mode deformation of rapidly spinning neutron stars with differential rotation. We perform fully relativistic three-dimensional simulations of compact stars with M/R ≥ 0.1, where M is the total gravitational mass and R the equatorial circumferential radius. We adopt an adiabatic equation of state with adiabatic index Γ = 2. As in Newtonian theory, we find that stars above a critical value of β ≡ T/W (where T is the rotational kinetic energy and W the gravitational binding energy) are dynamically unstable to bar formation. For our adopted choices of stellar compaction and rotation profile, the critical value of β = β_dGR is ∼0.24-0.25, only slightly smaller than the well-known Newtonian value ∼0.27 for incompressible Maclaurin spheroids. The critical value depends only very weakly on the degree of differential rotation for the moderate range we surveyed. All unstable stars form bars on a dynamical timescale. Models with sufficiently large β subsequently form spiral arms and eject mass, driving the remnant to a dynamically stable state. Models with moderately large β ≳ β_dGR do not develop spiral arms or eject mass but adjust to form dynamically stable ellipsoidal-like configurations. If the bar-mode instability is triggered in supernova collapse or binary neutron star mergers, it could be a strong and observable source of gravitational waves. We determine characteristic wave amplitudes and frequencies.