Ultra light bosonic dark matter and cosmic microwave background
  In this paper, we consider the hypothesis in which a species of ultra light
bosonic dark matter (ULBDM) with mass $m_{B}\sim 10^{-22}$ eV could be the
dominant dark matter (DM) in the Universe. As a first approach we work in the
context of kinetic theory, where ULBDM is described by the phase space
distribution function whose dynamics is dictated by the Boltzmann-Einstein
equations. We investigate the effects that this kind of dark matter imprints in
the acoustic peaks of the cosmic microwave background. We find that the effect
of the Bose-Einstein statistics is small, albeit perceptible, and is equivalent
to an increase of non-relativistic matter. It is stressed that in this
approach, the mass-to-temperature ratio necessary for ULBDM to be a plausible
DM candidate is about five orders of magnitude. We show that reionization is
also necessary and we address a range of consistent values for this model. We
find that the temperature of ULBDM is below the critical value, impliying that
Bose-Einstein condensation is inherent to the ULBDM paradigm.
