Velocity correlation tensor and substructure's statistics around density peaks

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We derive the conditional probability distribution of the peculiar velocity around a peak of given overdensity in a Gaussian density field. This distribution characterizes the shear field around local, isolated density peaks. We use the moments of the velocity distribution to study the spatial distribution of the escape velocity from protostructures forming in a standard, flat, scale--invariant CDM model. In the case of high (ν 2) peaks with filtering scales Rf 2 h⁻¹ Mpc, we find that at distances between 2 ÷4 Mpc h⁻¹ (or equivalently ≈ 1 ÷2 Rf) a sensitive fraction of the matter enclosed within an extrapolated (up to the present epoch) volume containing a total gravitating mass M ≈ 2.725×10¹⁵ h⁻¹ M (of the same order as that derived from gravitational lensing observations, see Tyson and Fisher tf95) is gravitationally unbound to the peak. The fraction of unbound matter varies between approximately between 0.05-0.2 depending on the central height ν of the density peak, on the filtering radius and on the adopted normalization for the fluctuation spectrum. We show how these results support our previous conclusions regarding the delay of the collapse induced by the presence of small-scale substructure in clusters of galaxies formed in standard CDM cosmological models.