Neutrino oscillations, Lorentz=CPT violation, and dark energy

If dark energy (DE) couples to neutrinos, then there may be apparent violations of Lorentz=CPT invariance in neutrino oscillations. The DE-induced Lorentz=CPT violation takes a specific form that introduces neutrino oscillations that are energy independent, differ for particles and antiparticles, and can lead to novel effects for neutrinos propagating through matter. We show that ultra-high-energy neutrinos may provide one avenue to seek this type of Lorentz=CPT violation in oscillations, improving the current sensitivity to such effects by 7 orders of magnitude. Lorentz=CPT violation in electron-neutrino oscillations may be probed with the zenith-angle dependence for high-energy atmospheric neutrinos. More compelling evidence for a DE-neutrino coupling would be provided by a dependence of neutrino oscillations on the direction of the neutrino momentum relative to our peculiar velocity with respect to the cosmic microwave background rest frame. While the amplitude of this directional dependence is expected to be small, it may nevertheless be worth seeking in current data and may be a target for future neutrino experiments.