  Direct measurements of the core-collapse supernova rate in the redshift range
0<z 1 appear to be about a factor of two smaller than the rate inferred from
the measured cosmic massive-star formation rate (SFR). We explore the
possibility that one could clarify the source of this "supernova rate problem"
by detecting the energy spectrum of supernova relic neutrinos with a next
generation detector like Hyper-Kamiokande. We make an alternative compilation
of the SFR data. We show that by only including published SFR data for which
the dust obscuration has been directly determined, the ratio of the observed
massive SFR to the observed supernova rate has large uncertainties$\sim
1.8^{+1.6}_{-0.6}$, and is statistically consistent with no supernova rate
problem. If we consider that a significant fraction of massive stars end their
lives as faint ONeMg SNe or as failed SNe, then the ratio reduces to $\sim
1.1^{+1.0}_{-0.4}$ and the rate problem is solved. We study the sources of
uncertainty involved in estimates of the neutrino detection rate and analyze
whether the spectrum of relic neutrinos can be used to independently identify
the existence of a supernova rate problem and its source. We consider an
ensemble of published and unpublished neutrino luminosities and temperatures
from core collapse supernova simulation models. We illustrate how the spectrum
of detector events might be used to constrain the average neutrino temperature
and SN models. We study the effects of neutrino oscillations on the detected
neutrino spectrum and also analyze a possible enhanced contribution from failed
supernovae. We conclude that it might be possible to measure the neutrino
temperature, neutrino oscillations, the EOS, and confirm this source of missing
luminous supernovae.
