How big were the first cosmological objects?

We calculate the cooling times at constant density for halos with virial temperatures from 100 K to 10^5 K that originate from a 3-sigma fluctuation of a CDM power spectrum in three different cosmologies. Our intention is to determine the first objects that can cool to low temperatures, but not to follow their dynamical evolution. We identify two generations of halos: those with low virial temperatures, Tvir < 9000 K that remain largely neutral, and those with larger virial temperatures that become ionized. The lower-temperature, lower-mass halos are the first to cool to 75 percent of their virial temperature. The precise temperature and mass of the first objects are dependent upon the molecular hydrogen (H2) cooling function and the cosmological model. The higher-mass halos collapse later but, in this paradigm, cool much more efficiently once they have done so, first via electronic transitions and then via molecular cooling: in fact, a greater residual ionization once the halos cool below 9000 K results in an enhanced H2 production and hence a higher cooling rate at low temperatures than for the lower-mass halos, so that within our constant-density model it is the former that are the first to cool to really low temperatures. We discuss the possible significance of this result in the context of CDM models in which the shallow slope of the initial fluctuation spectrum on small scales leads to a wide range of halo masses (of differing overdensities) collapsing over a small redshift interval. This ``crosstalk'' is sufficiently important that both high- and low-mass halos collapse during the lifetimes of the massive stars which may be formed at these epochs. Further investigation is thus required to determine which generation of halos plays the dominant role in early structure formation.