Effects of a neutrino-dark energy coupling on oscillations of
  high-energy neutrinos
  If dark energy (DE) is a dynamical field rather than a cosmological constant,
an interaction between DE and the neutrino sector could exist, modifying the
neutrino oscillation phenomenology and causing CP and apparent Lorentz
violating effects. The terms in the Hamiltonian for flavor propagation induced
by the DE-neutrino coupling do not depend on the neutrino energy, while the
ordinary components decrease as $\Delta m^2/E_{\nu}$. Therefore, the DE-induced
effects are absent at lower neutrino energies, but become significant at higher
energies, allowing to be searched for by neutrino observatories. We explore the
impact of the DE-neutrino coupling on the oscillation probability and the
flavor transition in the three-flavor framework, and investigate the
CP-violating and apparent Lorentz violating effects. We find that DE-induced
effects become observable for $E_{\nu}m_{\text{eff}} \sim 10^{-20}~
\text{GeV}^2$, where $m_{\rm eff}$ is the effective mass parameter in the
DE-induced oscillation probability, and CP is violated over a wide energy
range. We also show that current and future experiments have the sensitivity to
detect anomalous effects induced by a DE-neutrino coupling and probe the new
mixing parameters. The DE-induced effects on neutrino oscillation can be
distinguished from other new physics possibilities with similar effects,
through the detection of the directional dependence of the interaction, which
is specific to this interaction with DE. However, current experiments will not
yet be able to measure the small changes of $\sim 0.03\%$ in the flavor
composition due to this directional effect.
