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.