  Using the formalism of the spherical infall model the structure of collapsed
and virialized dark halos is calculated for a variety of scale-free initial
conditions. In spite of the scale-free cosmological nature of the problem, the
collapse of individual objects is not self-similar. Unlike most of previous
calculations the dynamics used here relies only on adiabatic invariants and not
on self-similarity. The paper focuses on the structure of the innermost part of
the collapsed halos and addresses the problem of central density cusps. The
slopes of density profiles at 1% of virial radius are calculated for a variety
of cosmological models and are found to vary with the mass of the halos and
power spectrum of the initial conditions. The inner slopes range between r^-2.3
and r^-2 with the limiting case of r^-2 reached for the largest masses. The
steep cusps found here correspond to the limiting case where all particles move
on radial orbits. The introduction of angular momentum will make the density
profile shallower. We expect this to resolve the discrepancy found between the
calculated profiles and the ones found in high resolution N-body simulations,
where the exponent ranges from -0.5 to -1.5. The robust prediction here is that
collisionless gravitational collapse in an expanding universe is expected to
form density cups and not halos with a core structure.
