  The spectrum of the supernova relic neutrino background (SRN) from
collapse-driven supernovae ever occurred in the universe is calculated by using
a realistic, time-dependent supernova rate derived from a standard model of
galaxy evolution based on the population synthesis method. The SRN spectrum we
show here is the most realistic at present, because the largest uncertainty in
previous theoretical predictions has come from unrealistic assumptions of the
supernova rate so far made. The SRN is one of the targets of the
Superkamiokande (SK) detector which will be constructed in a year and the SRN,
if at all detected, would provide a new tool to probe the history of supernova
explosions in the universe. The expected event rate at the SK is therefore
calculated in this paper. Our major results include: (1) the supernova rate is
much higher in the early phase of evolution of galaxies and there appears a
hump in the SRN spectrum in the low-energy region of $\ltilde 5$ MeV, (2) the
SRN flux depends on the Hubble constant ($H_0$) in a way approximately
proportional to $H_0^2$ and only weakly on the density parameter of the
universe ($\Omega_0$) and a cosmological constant ($\lambda_0$), and (3) the
plausible event rate at the SK is 1.2 yr$^{-1}$ in the observable energy range.
Such a low event rate is due mainly to a quite low supernova rate at present
which is averaged over the morphological types of galaxies. The most optimistic
rate in our model is found to be 4.7 yr$^{-1}$, and if more events are
detected, we will have to reconsider our current understanding of
collapse-driven supernovae and evolution of galaxies.
