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I address the question of what can be learned from the observation of thediffuse supernova neutrino flux in the precision phase, at next generationdetectors of Megaton scale. An analytical study of the spectrum of the diffuseflux shows that, above realistic detection thresholds of 10 MeV or higher, thespectrum essentially reflects the exponential-times-polynomial structure of theoriginal neutrino spectrum at the emission point. There is only a weak (tens ofper cent) dependence on the power \beta describing the growth of the supernovarate with the redshift. Different original neutrino spectra correspond to largedifferences in the observed spectrum of events at a water Cerenkov detector:for typical supernova rates, the ratio of the numbers of events in the firstand second energy bins (of 5 MeV width) varies in the interval 1.5 - 4.3 forpure water (energy threshold 18 MeV) and in the range 1 - 2.5 for water withGadolinium (10 MeV threshold). In the first case discrimination would bedifficult due to the large errors associated with background. With Gadolinium,instead, the reduction of the total error down to 10-20 % level would allowspectral sensitivity, with a dramatic improvement of precision with respect tothe SN1987A data. Even in this latter case, for typical neutrino luminosity thedependence on \beta is below sensitivity, so that it can be safely neglected indata analysis.Comment: LaTeX, 10 pages, 5 figures; details added to fig. 5 and related text,  minor modifications to the text, references added. Version in press in  Phys.Rev.D

Testing neutrino spectra formation in collapsing stars with the diffuse supernova neutrino flux