Astrophysical Limits on Superluminal Electron and Neutrino Velocities

The observation of two PeV-scale neutrino events reported by Ice Cube allows one to place constraints on Lorentz invariance violation (LIV) in the,neutrino sector. First, I derive an upper limit for \textit{the difference} between putative superluminal neutrino and electron velocities of \textless\ $\sim$5.6 x 10$^{-19}$ in units where c $=$ 1. I then derive a new, strong constraint on superluminal electron velocitiy $\delta_{\mathrm{e}}$ \textless\ 5 x 10$^{-21}$. One then obtains an upper limit on the superluminal neutrino velocity \textit{alone} of $\delta \nu $ \textless\ 5.6 x 10$^{-19}$, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are \textit{subluminal} the constraint on \textbar $\delta_{\mathrm{e}}$\textbar \textless\ 8 x 10$^{-17}$, obtained from the Crab Nebula $\gamma $-ray spectrum, places a weaker constraint on superluminal neutrino velocity of \textless\ 8 $\times$ 10$^{-17}$.