  It is greatly expected that the relic neutrino background from past
supernovae is detected by Superkamiokande (SK) which is now under construction.
We calculate the spectrum and the event rate at SK systematically by using the
results of simulations of a supernova explosion and reasonable supernova rates.
We also investigate the effect of a cosmological constant, $\Lambda$, on the
spectrum, since some recent cosmological observations strongly suggest the
existence of $\Lambda$. We find following results. 1) The spectrum has a peak
at about 3 MeV, which is much lower than that of previous estimate ($6 \sim$ 10
MeV). 2) The event rate at SK in the range from 10 MeV to 50 MeV, where the
relic neutrinos from past supernovae is dominant, is about $25 \, {h_{50}}^2 \,
\left( \frac{R_{SN}}{0.1 {\rm yr^{-1}}} \right) \left(\frac{n_G \,
h_{50}^{-3}}{0.02{\rm Mpc}^{-3}}\right)$ events per year, where $R_{SN}$ is the
supernova rate in a galaxy, $n_G$ is the number density of galaxies, and
$h_{50} = H_0 /$ (50km/s/Mpc), where $H_0$ is the Hubble constant. 3) The event
rate is almost insensitive to $\Lambda$. The flux increases in the low energy
side ($< 10$ MeV) with increasing $\Lambda$, but decreases in the high energy
side (10 MeV $<$) in models in which the integrated number of supernovae in one
galaxy is fixed.
