Radial mode stability of two-fluid neutron stars

Radial mode stability is a necessary condition for the astrophysical viability of compact objects. In recent years, astrophysical models with two fluids have gained popularity, especially in their ability to model dark matter admixed neutron stars. Just as is the case of single-fluid stars, a stability criterion based on the background equations has been developed - the critical curve for the particle numbers of the two fluids in the two-dimensional configuration space determines a one-dimensional sequence that labels the marginally stable configurations - but its validity depends on the linear stability of radial perturbations which remains unstudied. In this paper, we establish a set of stability criteria for two perfect-fluid relativistic stars by carefully studying the radial mode perturbation equations. We prove that modes are complete, have real eigenvalues with a minimum eigenvalue (i.e., a fundamental mode), thus a configuration is stable if and only if the fundamental mode is positive. As a consequence, our work formally and rigorously proves these necessary conditions for the stability criterion based on the background equations.