Time dependent CP -even and CP -odd signatures of scalar ultralight dark matter in neutrino oscillations

Scalar ultralight dark matter (ULDM) interacting with neutrinos can induce, under certain conditions, time dependent modifications to neutrino oscillation probabilities. The limit in which the ULDM perturbation can be treated as constant throughout the neutrino propagation time has been addressed by several previous works. We complement these by systematically analyzing the opposite limit - accounting for the temporal variations of the ULDM field by solving time dependent Schrödinger equations. In particular, we study a novel two-generations-like CP violating (CPV) signature unique to rapidly oscillating ULDM. We derive the leading order, time dependent corrections to the oscillation probabilities, for both CP conserving and CPV couplings, and explain how they can be measured in current and future experiments.