  The halo mass function, dn/dM, predicted by hierarchical clustering models
can be measured indirectly using dynamical probes like the distribution of
gravitational lens image separations, dn/dDelta-theta, or halo circular
velocities, dn/d v_c. These dynamical variables depend on the halo structure as
well as the halo mass. Since baryonic physics, particularly cooling,
significantly modifies the central density structure of dark matter halos, both
observational distributions show a feature corresponding to the mass scale
below which the baryons in the halo can cool (i.e. galaxies versus clusters).
We use simplified but self-consistent models to show that the structural
changes to the halos produced by the cooling baryons explain both
distributions. Given a fixed halo mass function, matching the observed image
separation distribution or local velocity function depends largely on Omega_b
through its effects on the cooling time scales. These baryonic effects on the
halo structure also affect the evolution of the velocity function of galaxies
with redshift.
