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 $F(\phi)R$, where $F$ depends on a scalar field
$\phi$ 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 $l \geq 2$ is ghost-free under the condition $F(\phi)>0$, with the
speed of gravity equivalent to that of light. For even-parity perturbations
with $l \geq 2$, 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 $F(\phi)>0$. As long as the perfect
fluid satisfies a weak energy condition with a positive propagation speed
squared $c_m^2$, there are neither ghost nor Laplacian instabilities for
theories of spontaneous scalarization and Brans-Dicke (BD) theories with a BD
parameter $\omega_{\rm BD}>-3/2$ (including $f(R)$ gravity). In these theories,
provided $0<c_m^2 \le 1$, we show that all the propagation speeds of
even-parity perturbations are sub-luminal inside the star, while the speeds of
gravity outside the star are equivalent to that of light.
