Cosmic Ray Spectrum and Tachyonic Neutrino

In the cosmic ray spectrum, there are two knees (abrupt changes of slopes) located around the energies $E_{th}=10^{15.5}$ eV and $10^{17.8}$ eV respectively. Based on the pioneering work by Kostelecky and Ehrlich, we ascribe the first knee to a sudden opening of the reaction channel $\tilde{\nu}_e+p\to n+e^+$ when the proton has a velocity just exceeding a critical value and so can absorb a tachyonic neutrino $\nu_e$ in the form of an antineutrino $\tilde{\nu}_e$. Similarly, the second knee is triggered by the reaction $\tilde{\nu}_\mu+p\to \Lambda+\mu^+$. The fitting of these two values of $E_{th}$ gives the tachyon mass of $\nu_e$ and $\nu_\mu$ being 0.54 eV/$c^2$ and 0.48 eV/$c^2$ respectively, which are in favor of a minimal three-flavor model for tachyonic neutrino with one parameter $\delta$ being estimated to be $\delta=0.34$ eV