Instability of compact stars with a nonminimal scalar-derivative coupling

Understand

For a theory in which a scalar field $\phi$ has a nonminimal derivative coupling to the Einstein tensor $G_{\mu \nu}$ of the form $\phi\,G_{\mu \nu}\nabla^{\mu}\nabla^{\nu} \phi$, it is known that there exists a branch of static and spherically-symmetric relativistic stars endowed with a scalar hair in their interiors.

• We study the stability of such hairy solutions with a radial field dependence $\phi(r)$ against odd- and even-parity perturbations.
• We show that, for the star compactness ${\cal C}$ smaller than $1/3$, they are prone to Laplacian instabilities of the even-parity perturbation associated with the scalar-field propagation along an angular direction.
• Even for ${\cal C}>1/3$, the hairy star solutions are subject to ghost instabilities.