Einstein static universe

The Einstein static universe has a time-independent scale factor and positively curved spatial sections with the geometry of a round three-sphere. For pressureless matter and a positive cosmological constant, its static conditions in units $c=1$ are $\Lambda =4\pi G\rho$ and $a_E^{-2}=\Lambda$.

The first equality balances matter's attractive contribution in the Friedmann acceleration equation against the cosmological repulsion; the second follows from the Friedmann equation with zero expansion. Spatial curvature enters that constraint, rather than being the attractive source in the acceleration balance. This is an exact equilibrium model. Whether a perturbed model remains near it requires a separate stability analysis.