Einstein static universes are unstable in generic $f(R)$ models

We study Einstein static universes in the context of generic $f(R)$ models. It is shown that Einstein static solutions exist for a wide variety of modified gravity models sourced by a barotropic perfect fluid with equation of state $w=p/\rho$, but these solutions are always unstable to either homogeneous or inhomogeneous perturbations. Our general results are in agreement with specific models investigated in that past. We also discuss how our techniques can be applied to other scenarios in $f(R)$ gravity.

Figure 1

Regions of stability in the $(\alpha ,w)$ parameter space for homogeneous perturbations of Einstein static universes. Note that there exist unstable $k=0$ modes for all models with $w>0$.

Figure 2

Regions of stability in the $(\alpha ,w)$ parameter space for $k^2=8$ ($n=2$) perturbations of Einstein static universes. Note that all models with $w<0$ are unstable with respect to these types of perturbations.