Stability of relativistic stars with scalar hairs

TL;DR

This paper analyzes the stability of relativistic stars in scalar-tensor theories with scalar hairs, deriving conditions for ghost-free and stable perturbations, and confirming stability and sub-luminal propagation speeds in specific models.

Contribution

It provides a comprehensive stability analysis of hairy relativistic stars in scalar-tensor theories, including no-ghost conditions and propagation speeds for various perturbations.

Findings

Odd-parity perturbations are ghost-free if F(φ)>0.

All perturbation speeds are sub-luminal inside the star for certain conditions.

Theories like spontaneous scalarization and Brans-Dicke with ω_{BD}>-3/2 are stable under these criteria.

Abstract

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.