We reconsider the collapse of cosmic structures in an Einstein-de Sitter universe, using the self-similar initial conditions of Fillmore & Goldreich (1984). We first derive a new approximation to describe the dark matter dynamics in spherical geometry, which we refer to as the "fluid approach." This method enables us to recover the self-similarity solutions of Fillmore and Goldreich for dark matter. We also derive new self-similarity solutions for the gas. We thus compare directly gas and dark matter dynamics, focusing on the differences due to their different dimensionalities in velocity space. This work may have interesting consequences for gas and dark matter distributions in large galaxy clusters, allowing us to explain why the total mass profile is always steeper than the X-ray gas profile. We also discuss the shape of the dark matter density profile found in N-body simulations in terms of a change of dimensionality in the dark matter velocity space. The stable clustering hypothesis has been finally considered in light of this analytical approach.
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