Instability of compact stars with a nonminimal scalar-derivative coupling

For a theory in which a scalar field $ϕ$ has a nonminimal derivative coupling to the Einstein tensor $G_{μν}$ of the form $ϕ\,G_{μν}\nabla^μ\nabla^ν ϕ$, 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 $ϕ(r)$ 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. We also find that even the other branch with a vanishing background field derivative is unstable for a positive perfect-fluid pressure, due to nonstandard propagation of the field perturbation $δϕ$ inside the star. Thus, there are no stable star configurations in derivative coupling theory without a standard kinetic term, including both relativistic and nonrelativistic compact objects.

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