Analytical relations concerning the collapse time in hierarchically clustered cosmological models

By means of numerical methods, we solve the equations of motion for the collapse of a shell of baryonic matter, made of galaxies and substructure falling into the central regions of a cluster of galaxies, taking into account the effect of the dynamical friction. The parameters on which the dynamical friction mainly depends are: the peaks’ height, the number of peaks inside a protocluster multiplied by the correlation function evaluated at the origin, the filtering radius and the nucleus radius of the protocluster of galaxies. We show how the collapse time τ of the shell depends on these parameters. We give a formula that links the dynamical friction coefficient η to the parameters mentioned above and an analytic relation between the collapse time and η. Finally, we obtain an analytical relation between τ and the mean overdensity δ within the shell. All the analytical relations that we find are in excellent agreement with the numerical integration.