Structure and History of Dark Matter Halos Probed with Gravitational
  Lensing
  We test with gravitational lensing data the dark matter (DM) halos embedding
the luminous baryonic component of galaxy clusters; our benchmark is provided
by their two-stage cosmogonical development that we compute with its variance,
and by the related '\alpha-profiles' we derive. The latter solve the Jeans
equation for the self-gravitating, anisotropic DM equilibria, and yield the
radial runs of the density \rho(r) and the velocity dispersion \sigma_r^2(r) in
terms of the DM 'entropy' K = \sigma_r^2/\rho^(2/3) ~ r^(\alpha) highlighted by
recent N-body simulations; the former constrains the slope to the narrow range
\alpha ~ 1.25 - 1.3. These physically based \alpha-profiles meet the overall
requirements from gravitational lensing observations, being intrinsically
flatter at the center and steeper in the outskirts relative to the empirical
NFW formula. Specifically, we project them along the l.o.s. and compare with a
recent extensive dataset from strong and weak lensing observations in and
around the cluster A1689. We find an optimal fit at both small and large scales
in terms of a halo constituted by an early body with \alpha ~ 1.25 and by
recent extensive outskirts, that make up an overall mass 10^15 M_sun with a
concentration parameter c ~ 10 consistent with the variance we compute in the
\LambdaCDM cosmogony. The resulting structure corresponds to a potential well
shallow in the outskirts as that inferred from the X rays radiated from the hot
electrons and baryons constituting the intracluster plasma.
