Stability of the Einstein static universe in modified Gauss-Bonnet gravity

We analyze the stability of the Einstein static universe by considering linear homogeneous perturbations in the context of $f(\mathcal{G})$ modified Gauss-Bonnet theories of gravity. By considering a generic form of $f(\mathcal{G})$, the stability region of the Einstein static universe is parametrized by the linear equation of state parameter $w=p/\rho$ and the second derivative $f^{\prime \prime }(\mathcal{G})$ of the Gauss-Bonnet term.

Figure 1

Regions of stability in the $(w,\alpha )$ parameter space for homogeneous perturbations of Einstein static universes. The left section is the phantom region, the center section is the general relativistic analogue region, and the right section corresponds to a normal matter region. Note that there exist stable modes for all $w$, provided $\alpha$ is chosen appropriately.