Zero average values of cosmological perturbations as an indispensable condition for the theory and simulations.

We point out a weak side of the commonly used determination of scalar cosmological perturbations lying in the fact that their average values can be nonzero for some matter distributions. It is shown that introduction of the finite-range gravitational potential instead of the infinite-range one resolves this problem. The concrete illustrative density profile is investigated in detail in this connection.

Lattice Universe: examples and problems.

We consider lattice Universes with spatial topologies [Formula: see text], [Formula: see text], and [Formula: see text]. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the [Formula: see text] model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.

Rigorous theoretical constraint on constant negative EoS parameter [Formula: see text] and its effect for the late Universe.

In this paper, we consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies). Supposing that the Universe contains also the cosmological constant and a perfect fluid with a negative constant equation of state (EoS) parameter [Formula: see text] (e.