  The first luminous objects in the concordance cosmology form by molecular
hydrogen cooling in dark matter dominated halos of masses ~10^6 M_sun. We use
Eulerian adaptive mesh refinement simulations to demonstrate that in the
presence of a large soft ultraviolet radiation background, molecular hydrogen
is the dominant coolant. Even for very large radiation backgrounds, the halo
masses that cool and collapse are up to two orders of magnitude smaller than
the halos that cool via atomic hydrogen line cooling. The abundance of cooling
halos and the cosmic mass fraction contained within them depends exponentially
on this critical mass scale. Consequently, the majority of current models of
cosmological reionization, chemical evolution, supermassive black hole
formation, and galaxy formation underestimate the number of star forming
progenitors of a given system by orders of magnitude. At the highest redshifts,
this disagreement is largest. We also show that even in the absence of residual
electrons, collisional ionization in central shocks create a sufficient amount
of electrons to form molecular hydrogen and cool the gas in halos of virial
temperatures far below the atomic cooling limit.
