Gamma ray bursts as probes of neutrino mass, quantum gravity, and dark energy

We calculate the time delays of neutrinos emitted in gamma ray bursts due to the effects of neutrino mass and quantum gravity using a time dependent Hubble constant which can significantly change the naive results presented hitherto in the literature for large redshifts, and gives some sensitivity to the details of dark energy. We show that the effects of neutrino mass, quantum gravity and dark energy may be disentangled by using low energy neutrinos to study neutrino mass, high energy neutrinos to study quantum gravity, and large redshifts to study dark energy. From low energy neutrinos one may obtain direct limits on neutrino masses of order ${10}^{-3}\mathrm{eV},$ and distinguish a neutrino mass hierarchy from an inverted mass hierarchy. From ultra-high energy neutrinos the sensitivity to the scale of quantum gravity can be pushed up to $E_{\mathrm{QG}}\sim 5\times {10}^{30}\mathrm{GeV}.$ By studying neutrinos from GRBs at large redshifts a cosmological constant could be distinguished from quintessence.