How small were the first cosmological objects?

The minimum mass that a virialized gas cloud must have in order to be able to cool in a Hubble time is computed, using a detailed treatment of the chemistry of molecular hydrogen. With a simple model for halo profiles, we reduce the problem to that of numerically integrating a system of chemical equations. The results agree well with numerically expensive three-dimensional simulations, and our approach has the advantage of being able to explore large regions of parameter space rapidly. The minimum baryonic mass M(b) is found to be strongly redshift dependent, dropping from 10(6) M. at z similar to 15 to 5 x 10(3) M. at z similar to 100 as molecular cooling becomes effective. For z much greater than 100, M(b) rises again, as cosmic microwave background photons inhibit H-2 formation through the H- channel. Finally, for z much greater than 200, the H-2(+) channel for H-2 formation becomes effective, driving M(b) down toward M(b) similar to 10(3) M.. With a standard cold dark matter power spectrum with sigma(8) = 0.7, this implies that a fraction 10(-3) of all baryons may have formed luminous objects by z = 30, which could be sufficient to reheat the universe.