Refractive neutrino masses, ultralight dark matter and cosmology

We consider in detail a possibility that the observed neutrino oscillations are due to refraction on ultralight scalar boson dark matter. We introduce the refractive mass squared, $\tilde{m}^2$, and study its properties: dependence on neutrino energy, state of the background, etc. If the background is in a state of cold gas of particles, $\tilde{m}^2$ shows a resonance dependence on energy. Above the resonance ($E \gg E_R $), we find that $\tilde{m}^2$ has the same properties as usual vacuum mass squared. Below the resonance, $\tilde{m}^2$ decreases with energy, which (if realised) allows us to avoid the cosmological bound on the sum of neutrino masses. Also, $\tilde{m}^2$ may depend on time. We consider the validity of the results: effects of multiple interactions with scalars, and modification of the dispersion relation. We show that for values of parameters of the system required to reproduce the observed neutrino masses, perturbativity is broken at low energies, which border above the resonance. If the background is in the state of coherent classical field, the refractive mass does not depend on energy explicitly but may show time dependence. It coincides with the refractive mass in a cold gas at high energies. The refractive nature of neutrino mass can be tested by searches of its dependence on energy and time.