Holes in the static Einstein universe and the model of the cosmological voids

Spherically symmetric, static model of the cosmological voids is constructed in the framework of the Tolman-Oppenheimer-Volkov equation with the cosmological constant. Extension of the Tooper result (dimensionless form of the TOV equation) is provided for non-zero $\Lambda$. Then, the equation is simplified in $\alpha \to 0$, $\lambda \to 0$, $\lambda/\alpha = const$ regime, suitable for largest structures in $\Lambda$-dominated universe. Voids are treated as an underdensity regions in the static Einstein universe. Both overdensity and underdensity (relative to static universe) solutions exist. They are identified with standard astrophysical spherical objects and voids, respectively. Model is tested against observed properties (the radius - the central density relation) and density profiles of voids. Analytical formulae for radial density contrast profile and radii of the voids are derived. Some consequences for cosmological n-body simulations are suggested. Hints on the dark matter/dark energy EOS filling the voids are provided.