  It has been hypothesized that the cosmic microwave background (CMB) provides
a temperature floor for collapsing protostars that can regulate the process of
star formation and result in a top-heavy initial mass function at high
metallicity and high redshift. We examine whether this hypothesis has any
testable observational consequences. First we determine, using a set of
hydrodynamic galaxy formation simulations, that the CMB temperature floor would
have influenced the majority of stars formed at redshifts between z=3 and 6,
and probably even to higher redshift. Five signatures of CMB-regulated star
formation are: (1) a higher supernova rate than currently predicted at high
redshift; (2) a systematic discrepancy between direct and indirect measurements
of the high redshift star formation rate; (3) a lack of surviving globular
clusters that formed at high metallicity and high redshift; (4) a more rapid
rise in the metallicity of cosmic gas than is predicted by current simulations;
and (5) an enhancement in the abundances of alpha elements such as O and Mg at
metallicities -2 < [Fe/H] < -0.5. Observations are not presently able to either
confirm or rule out the presence of these signatures. However, if correct, the
top-heavy IMF of high-redshift high-metallicity globular clusters could provide
an explanation for the observed bimodality of their metallicity distribution.
