Static cosmological solutions in quadratic gravity

TL;DR

This paper investigates the existence and stability of static cosmological solutions in quadratic gravity, revealing conditions under which such solutions are stable or unstable depending on curvature, matter content, and perturbations.

Contribution

It provides a detailed analysis of static solutions in quadratic gravity, including stability criteria for both isotropic and anisotropic perturbations, and identifies conditions for stability without violating energy conditions.

Findings

Negative curvature static solutions are always unstable with positive energy density.

Positive curvature static solutions can be stable under specific conditions.

Stability with anisotropic perturbations imposes additional restrictions.

Abstract

We consider conditions for existence and stability of a static cosmological solution in quadratic gravity. It appears that such a solution for a Universe filled by only one type of perfect fluid is possible in a wide range of the equation of state parameter $w$ and for both positively and negatively spatially curved Universe. We show that the static solution for the negative curvature is always unstable if we require positive energy density of the matter content. On the other hand, a static solution with positive spatial curvature can be stable under certain restrictions. Stability of this solution with respect to isotropic perturbation requires that the coupling constant with the $R^2$ therm in the Lagrangian of the theory is positive, and the equations of state parameter $w$ is located in a rather narrow interval. Nevertheless, the stability condition does not require violation of the Strong Energy Condition. Taking into account anisotropic perturbations leads to further restrictions on the values of coupling constants and the parameter $w$.