The Existence of Einstein Static Universes and their Stability in Fourth order Theories of Gravity

TL;DR

This paper examines the existence and stability of Einstein Static universes within $f(R)$ gravity, finding that only specific models admit such universes and analyzing their stability under various perturbations.

Contribution

It identifies the class of $f(R)$ theories that admit Einstein Static universes and analyzes their stability properties under different perturbations.

Findings

Only one class of $f(R)$ theories admits Einstein Static universes.

These universes are neutrally stable against vector and tensor perturbations.

Scalar perturbations are stable if the matter fluid's sound speed exceeds approximately 0.21.

Abstract

We investigate whether or not an Einstein Static universe is a solution to the cosmological equations in $f(R)$ gravity. It is found that only one class of $f(R)$ theories admits an Einstein Static model, and that this class is neutrally stable with respect to vector and tensor perturbations for all equations of state on all scales. Scalar perturbations are only stable on all scales if the matter fluid equation of state satisfies $c_s^2>\frac{\sqrt{5}-1}{6}\approx 0.21$. This result is remarkably similar to the GR case, where it was found that the Einstein Static model is stable for $c_s^2>{1/5}$.