Limits on the Neutrino Velocity, Lorentz Invariance, and the Weak   Equivalence Principle with TeV Neutrinos from Gamma-Ray Bursts
  Five TeV neutrino events weakly correlated with five gamma-ray bursts (GRBs) were detected recently by IceCube. This work is an attempt to show that if the GRB identifications are verified, the observed time delays between the TeV neutrinos and gamma-ray photons from GRBs provide attractive candidates for testing fundamental physics with high accuracy. Based on the assumed associations between the TeV neutrinos and GRBs, we find that the limiting velocity of the neutrinos is equal to that of photons to an accuracy of $\sim1.9\times10^{-15}-2.5\times10^{-18}$, which is about $10^{4}-10^{7}$ times better than the constraint obtained with the neutrino possibly from a blazar flare. In addition, we set the most stringent limits up to date on the energy scale of quantum gravity for both the linear and quadratic violations of Lorentz invariance, namely $E_{\rm QG, 1}>6.3\times10^{18}-1.5\times10^{21}$ GeV and $E_{\rm QG, 2}>2.0\times10^{11}-4.2\times10^{12}$ GeV, which are essentially as good as or are an improvement of one order of magnitude over the results previously obtained by the GeV photons of GRB 090510 and the PeV neutrino from a blazar flare. Assuming that the Shapiro time delay is caused by the gravitational potential of the Laniakea supercluster of galaxies, we also place the tightest limits to date on Einstein's weak equivalence principle through the relative differential variations of the parameterized post-Newtonian parameter $\gamma$ values for two different species of particles (i.e., neutrinos and photons), yielding $\Delta\gamma \sim 10^{-11}-10^{-13}$. However, it should be emphasized again that these limits here obtained are at best forecast of what could be achieved if the GRB/neutrino correlations would be finally confirmed. 
Jun-Jie Wei, Xue-Feng Wu, He Gao, Peter M\'esz\'aros
astro-ph.HE gr-qc hep-ph