Relativistic star perturbations in Horndeski theories with a gauge-ready formulation

TL;DR

This paper develops a comprehensive gauge-ready framework for analyzing relativistic star perturbations within full Horndeski theories, enabling stability and quasi-normal mode studies of compact objects without fixing gauges.

Contribution

It introduces a versatile, gauge-independent formulation for perturbations in Horndeski theories, including matter, applicable to various astrophysical scenarios.

Findings

Derived second-order actions for odd- and even-parity perturbations without gauge fixing.

Identified propagation speeds of dynamical perturbations for high modes.

Established stability conditions applicable to neutron stars and black holes.

Abstract

We present a general framework for studying the relativistic star perturbations on a static and spherically symmetric background in full Horndeski theories. We take a perfect fluid into account as a form of the Schutz-Sorkin action. Our formulation is sufficiently versatile in that the second-order actions of perturbations in odd- and even-parity sectors are derived without choosing particular gauge conditions, so they can be used for any convenient gauges at hand. The odd-parity sector contains one dynamical gravitational degree of freedom coupled to a time-independent four velocity of the fluid. In the even-parity sector there are three dynamical perturbations associated with gravity, scalar field, and matter sectors, whose equations of motion are decoupled from other nondynamical perturbations. For high radial and angular momentum modes, we obtain the propagation speeds of all dynamical perturbations and show that the perfect fluid in the even-parity sector has a standard sound speed affected by neither gravity nor the scalar field. Our general stability conditions and perturbation equations of motion can be directly applied to the stabilities of neutron stars and black holes as well as the calculations of their quasi-normal frequencies.