The neutrino-nucleon cross section at UHE and its astrophysical implications

We present a quantitative study of the $\nu N$ cross section in the neutrino energy range $10^4<E_{\nu}<10^{14}$ GeV within two transversal QCD approaches: NLO DGLAP evolution using different sets of PDFs and BK small-$x$ evolution with running coupling and kinematical corrections. We show that the non-linear effects embodied in the BK equation yield a slower raise in the cross section for $E_{\nu}\gtrsim 10^{8}$ GeV than the usual DGLAP based calculation. Finally, we translate this theoretical uncertainty into upper bounds for the ultra-high-energy neutrino flux for different experiments.