  I address the question of what can be learned from the observation of the
diffuse supernova neutrino flux in the precision phase, at next generation
detectors of Megaton scale. An analytical study of the spectrum of the diffuse
flux shows that, above realistic detection thresholds of 10 MeV or higher, the
spectrum essentially reflects the exponential-times-polynomial structure of the
original neutrino spectrum at the emission point. There is only a weak (tens of
per cent) dependence on the power \beta describing the growth of the supernova
rate with the redshift. Different original neutrino spectra correspond to large
differences in the observed spectrum of events at a water Cerenkov detector:
for typical supernova rates, the ratio of the numbers of events in the first
and second energy bins (of 5 MeV width) varies in the interval 1.5 - 4.3 for
pure water (energy threshold 18 MeV) and in the range 1 - 2.5 for water with
Gadolinium (10 MeV threshold). In the first case discrimination would be
difficult due to the large errors associated with background. With Gadolinium,
instead, the reduction of the total error down to 10-20 % level would allow
spectral sensitivity, with a dramatic improvement of precision with respect to
the SN1987A data. Even in this latter case, for typical neutrino luminosity the
dependence on \beta is below sensitivity, so that it can be safely neglected in
data analysis.
