I study what can be learned from the observed diffuse supernova neutrino spectrum in the precision phase, at next generation detectors of Megaton scale. Different original neutrino spectra correspond to large differences in the observed spectrum of events at a water Cerenkov detector. For pure water 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 greater precision than the SN1987A data. The dependence on the power β of evolution of the supernova rate with redshift is below sensitivity.