Neutrino meets ultralight dark matter: $\boldsymbol{0\nu\beta\beta}$
  decay and cosmology
  We explore the neutrinoless double beta ($0\nu \beta\beta$) decay induced by
an ultralight dark matter field coupled to neutrinos. The effect on
$0\nu\beta\beta$ decay is significant if the coupling violates the lepton
number, for which the $\Delta L=2$ transition is directly driven by the dark
matter field without further suppression of small neutrino masses. As the
ultralight dark matter can be well described by a classical field, the effect
features a periodic modulation pattern in decay events. However, we find that
in the early Universe such coupling will be very likely to alter the standard
cosmological results. In particular, the requirement of neutrino free-streaming
before the matter-radiation equality severely constrains the parameter space,
such that the future $0\nu \beta\beta$ decay experiments can hardly see any
signal even with a meV sensitivity to the effective neutrino mass.
