  We use the redshift distribution of type-Ia supernovae (SNe) discovered by
the Supernova Cosmology Project to constrain the star formation history (SFH)
of the Universe and SN Ia progenitor models. Given some of the recent
determinations of the SFH, the observed SN Ia redshift distribution indicates a
long (>~1 h^-1 Gyr) mean delay time between the formation of a stellar
population and the explosion of some of its members as SNe Ia. For example, if
the Madau et al. (1998) SFH is assumed, the delay time tau is constrained to be
tau > 1.7 (tau > 0.7) h^-1 Gyr at the 95%(99%) confidence level (CL). SFHs that
rise at high redshift, similar to those advocated by Lanzetta et al. (2002),
are inconsistent with the data at the 95% CL unless tau > 2.5 h^-1 Gyr. Long
time delays disfavor progenitor models such as edge-lit detonation of a white
dwarf accreting from a giant donor, and the carbon core ignition of a white
dwarf passing the Chandrasekhar mass due to accretion from a subgiant. The SN
Ia delay may be shorter, thereby relaxing some of these constraints, if the
field star formation rate falls, between z=1 and the present, less sharply than
implied, e.g., by the original Madau plot. We show that the discovery of larger
samples of high-z SNe Ia by forthcoming observational projects should yield
strong constraints on the progenitor models and the SFH. In a companion paper
(astro-ph/0309797), we demonstrate that if SNe Ia produce most of the iron in
galaxy clusters, and the stars in clusters formed at z~2, the SN Ia delay time
must be lower than 2 Gyr. If so, then the Lanzetta et al. (2002) SFH will be
ruled out by the data presented here.
