Instability of compact stars with a nonminimal scalar-derivative coupling

TL;DR

This paper investigates the stability of scalar-hairy compact star solutions in a derivative coupling theory, revealing they are generally unstable due to Laplacian and ghost instabilities, with no stable configurations found.

Contribution

It demonstrates the instability of all scalar-hairy star solutions in a nonminimal derivative coupling theory, including both relativistic and nonrelativistic cases.

Findings

Stars with compactness less than 1/3 are Laplacian unstable.

Solutions with compactness greater than 1/3 are ghost unstable.

Even solutions with zero background field derivative are unstable for positive pressure.

Abstract

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. 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 $\delta \phi$ 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.