||We analytically study the evolution of gravitational instability of self-gravitating dark matter within the framework of a nonrelativistic “hydrodynamical” model of the Universe, valid for scales that are small compared to the Hubble scale and for distances far away from black holes. We propose a particular form for parametrization of the particle distribution function via macroscopic quantities, such that the initial dispersion of microscopic velocities is not neglected, but plays a determinant role. Thus our model may be called a modified cold dark matter model. We found an analytical solution which indicates that a spontaneous spatially localized fluctuation of velocity generates density perturbations relative to initially unperturbed background. For the instability to arise, we do not need to assume any initial density (metric) fluctuations. The evolving perturbation is hydrodynamically unstable in the self-gravitating expanding Universe and can produce both—regions where no dark matter accumulates and halolike regions where dark matter does accumulate. The perturbation region boundary propagates as a shock wave with a speed that is time varying, until eventually reaching its steady state. We also derive an explicit analytical expression for the correlation function R(x1-x2) of density fluctuations, which can be compared by experimentalists with data from astrophysical observations.