Dark matter halo's and self similarity
  This papers explores the self similar solutions of the Vlasov-Poisson system
and their relation to the gravitational collapse of dynamically cold systems.
Analytic solutions are derived for power law potential in one dimension, and
extensions of these solutions in three dimensions are proposed. Next the self
similarity of the collapse of cold dynamical systems is investigated
numerically. The fold system in phase space is consistent with analytic self
similar solutions, the solutions present all the proper self-similar scalings.
An additional point is the appearance of an $x^{-(1/2)}$ law at the center of
the system for initial conditions with power law index larger than $-(1/2)$. It
is found that the first appearance of the $x^{-(1/2)}$ law corresponds to the
formation of a singularity very close to the center. Finally the general
properties of self similar multi dimensional solutions near equilibrium are
investigated. Smooth and continuous self similar solutions have power law
behavior at equilibrium. However cold initial conditions result in
discontinuous phase space solutions, and the smoothed phase space density
looses its auto similar properties. This problem is easily solved by observing
that the probability distribution of the phase space density $P$ is identical
except for scaling parameters to the probability distribution of the smoothed
phase space density $P_S$. As a consequence $P_S$ inherit the self similar
properties of $P$. This particular property is at the origin of the universal
power law observed in numerical simulation for ${\rho}/{\sigma^3}$. The self
similar properties of $P_S$ implies that other quantities should have also an
universal power law behavior with predictable exponents. This hypothesis is
tested using a numerical model of the phase space density of cold dark matter
halo's, an excellent agreement is obtained.
