Stability of relativistic stars with scalar hairs

We study the stability of relativistic stars in scalar-tensor theories with a nonminimal coupling of the form F(φ)R, where F depends on a scalar field φ and R is the Ricci scalar. On a spherically symmetric and static background, we incorporate a perfect fluid minimally coupled to gravity as a form of the Schutz-Sorkin action. The odd-parity perturbation for the multipoles l≥2 is ghost-free under the condition F(φ)>0, with the speed of gravity equivalent to that of light. For even-parity perturbations with l≥2, there are three propagating degrees of freedom arising from the perfect-fluid, scalar-field, and gravity sectors. For l=0, 1, the dynamical degrees of freedom reduce to two modes. We derive no-ghost conditions and the propagation speeds of these perturbations and apply them to concrete theories of hairy relativistic stars with F(φ)>0. As long as the perfect fluid satisfies a weak energy condition with a positive propagation speed squared cm2, there are neither ghost nor Laplacian instabilities for theories of spontaneous scalarization and Brans-Dicke (BD) theories with a BD parameter ωBD>-3/2 (including f(R) gravity). In these theories, provided 0<cm2≤1, we show that all the propagation speeds of even-parity perturbations are subluminal inside the star, while the speeds of gravity outside the star are equivalent to that of light.