  From a cosmological perspective, scalar fields are well-motivated dark matter
and dark energy candidates. Several possibilities of neutrino couplings with a
time-varying cosmic field have been investigated in the literature. In this
work, we present a framework in which violations of Lorentz invariance (LIV)
and $CPT$ symmetry in the neutrino sector could arise from an interaction among
neutrinos with a time-varying scalar field. Furthermore, some cosmological and
phenomenological aspects and constraints concerning this type of interaction
are discussed. Potential violations of Lorentz and $CPT$ symmetries at present
and future neutrino oscillation experiments such as IceCube and KM3NeT can
probe this scenario.
