The IceCube telescope has detected diffuse neutrino emission , 20 events of which were reported to be above 60 TeV .
In this paper , we fit the diffuse neutrino spectrum using Poisson statistics , which are the most appropriate for the low counts per energy bin .
We extend the fitted energy range and exploit the fact that no neutrinos were detected above 2 PeV , despite the high detector sensitivity around the Glashow resonance at 6.3 PeV and beyond .
A best-fit power-law slope of \alpha = 2.9 \pm 0.3 is found with no evidence for a high-energy cutoff .
This slope is steeper than \alpha = 2.3 \pm 0.3 found by the IceCube team using a different fitting method .
Such a steep spectrum facilitates the identification of high energy ( \gg PeV ) neutrinos , if detected , to be due to the GZK effect of cosmic-ray protons interacting with the Extragalactic Background Light .
We use the ratio of EeV to PeV neutrinos in GZK models to show that the currently detected PeV neutrinos could not be due to the GZK effect , because this would imply many more higher-energy neutrinos that should have been detected , but were not .
The non-detection of GZK neutrinos by IceCube despite more than essentially 1200 observing days , has already ruled out ( at 95 % confidence ) models that predict rates of \sim 1 neutrino/yr or more .
We use this non-detection to quantify the confidence at which GZK models are ruled out , and compute the additional IceCube and ( in the future ) ARA observing time that would rule them out with 95 % confidence if no detection is made .