Gravitational equilibrium in the presence of a positive cosmological constant

We reconsider the virial theorem in the presence of a positive cosmological constant Λ. Assuming steady state, we derive an inequality of the form $\rho >~A(\Lambda /8\mathrm{}\pi G_N)$ for the mean density ρ of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at $A_{\mathrm{sphere}}=2,$ its value can increase by several orders of magnitude as the shape of the object deviates from a spherically symmetric one. This indicates that flattened matter distributions such as, e.g., clusters or superclusters, with low density, cannot be in gravitational equilibrium.