  Initially far out-of-equilibrium self-gravitating systems form, through a
collisionless relaxation dynamics, quasi-stationary states (QSS). These may
arise from a bottom-up aggregation of structures or in a top-down frame; their
quasi-equilibrium properties are well described by the Jeans equation and are
not universal, i.e. they depend on initial conditions. To understand the origin
of such dependence, we present results of numerical experiments of initially
cold and spherical systems characterized by various choices of the spectrum of
initial density fluctuations. The amplitude of such fluctuations determines
whether the system relaxes in a top-down or a bottom-up manner. We find that
statistical properties of the resulting QSS mainly depend upon the amount of
energy exchanged during the formation process. In particular, in the violent
top-down collapses the energy exchange is large and the QSS show an inner core
with an almost flat density profile and a quasi Maxwell-Boltzmann (isotropic)
velocity distribution, while their outer regions display a density profile
$\rho(r) \propto r^{-\alpha}$ ($\alpha >0$) with radially elongated orbits. We
analytically show that $\alpha=4$ in agreement with numerical experiments. In
the less violent bottom-up dynamics, the energy exchange is much smaller, the
orbits are less elongated and $0< \alpha(r) \le 4$, with a a density profile
well fitted by the Navarro-Frenk-White behavior. Such a dynamical evolution is
shown by both non-uniform spherical isolated systems and by halos extracted
from cosmological simulations. We consider the relation of these results with
the core-cusp problem concluding that this is naturally solved if galaxies form
through a monolithic collapse.
