Existence of Einstein static universes and their stability in fourth-order theories of gravity

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 $c_{\mathrm{s}}^2>\frac{\sqrt{5}-1}{6}\approx 0.21$. This result is remarkably similar to the general relativity case, where it was found that the Einstein static model is stable for $c_s^2>\frac{1}{5}$.