Stability of the Einstein static universe in $f(R)$ gravity

We analyze the stability of the Einstein static universe by considering homogeneous scalar perturbations in the context of $f(R)$ modified theories of gravity. By considering specific forms of $f(R)$, the stability regions of the solutions are parametrized by a linear equation of state parameter $w=p/\rho$. Contrary to classical general relativity, it is found that in $f(R)$ gravity a stable Einstein cosmos with a positive cosmological constant does indeed exist. Thus, we are lead to conclude that, in principle, modifications in $f(R)$ gravity stabilize solutions which are unstable in general relativity.

Figure 1

The stability regions, for the case of $f(R)\propto R+R^2$ theory, are depicted between the curves. The thick solid line represents general relativity. In the lower left region, which tends to general relativity, the cosmological constant is negative. However, in the upper triangularlike shaped region (AI, AII, and AIII) the Einstein static universe is stable and $\Lambda$ is positive; compare also with Fig. 2. Note that the equation of state parameter $w$ is strictly negative.

Figure 2

The stability regions, for the case of $f(R)\propto R+1/R$ theory, are depicted between the curves. As before, the thick solid line represents general relativity. In the lower left region, which tends to general relativity, the cosmological constant is negative. In the upper triangularlike shaped region (BI, BII, and BIII) the Einstein static universe is stable and $\Lambda$ is positive. As before, $w$ is strictly negative.