Spectrum of the relic neutrino background from past supernovae and cosmological models

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.