  Strong constraints on the cosmic star formation history (SFH) have recently
been established using ultraviolet and far-infrared measurements, refining the
results of numerous measurements over the past decade. Taken together, the most
recent and robust data indicate a compellingly consistent picture of the SFH
out to redshift z~6, with especially tight constraints for z<~1. We fit these
data with simple analytical forms, and derive conservative bands to indicate
possible variations from the best fits. Since the z<~1 SFH data are quite
precise, we investigate the sequence of assumptions and corrections that
together affect the SFH normalisation, to test their accuracy, both in this
redshift range and beyond. As lower limits on this normalisation, we consider
the evolution in stellar mass density, metal mass density, and supernova rate
density, finding it unlikely that the SFH normalisation is much lower than
indicated by our direct fit. Additionally, predictions from the SFH for
supernova type Ia rate densities tentatively suggests delay times of ~3 Gyr. As
a corresponding upper limit on the SFH normalisation, we consider the
Super-Kamiokande (SK) limit on the electron antineutrino flux from past
core-collapse supernovae, which applies primarily to z<~1. We find consistency
with the SFH only if the neutrino temperatures from SN events are relatively
modest. Constraints on the assumed initial mass function (IMF) also become
apparent. The traditional Salpeter IMF, assumed for convenience by many
authors, is known to be a poor representation at low stellar masses (<~ 1 solar
mass), and we show that recently favoured IMFs are also constrained. In
particular, somewhat shallow, or top-heavy, IMFs may be preferred, although
they cannot be too top-heavy. (Abridged)
