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 We consider an ensemble of published and unpublished neutrino luminosities and temperatures from core collapse supernova simulation models.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.
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