  Light scalars (as the axion) with mass m ~ 10^{-22} eV forming a
Bose-Einstein condensate (BEC) exhibit a Jeans length in the kpc scale and were
therefore proposed as dark matter (DM) candidates. Our treatment here is
generic, independent of the particle physics model and applies to all DM BEC,
in or out of equilibrium. Two observed quantities crucially constrain DM in an
inescapable way: the average DM density rho_{DM} and the phase-space density Q.
The observed values of rho_{DM} and Q in galaxies today constrain both the
possibility to form a BEC and the DM mass m. These two constraints robustly
exclude axion DM that decouples just after the QCD phase transition. Moreover,
the value m ~ 10^{-22} eV can only be obtained with a number of
ultrarelativistic degrees of freedom at decoupling in the trillions which is
impossible for decoupling in the radiation dominated era. In addition, we find
for the axion vacuum misalignment scenario that axions are produced strongly
out of thermal equilibrium and that the axion mass in such scenario turns to be
17 orders of magnitude too large to reproduce the observed galactic structures.
Moreover, we also consider inhomogenous gravitationally bounded BEC's supported
by the bosonic quantum pressure independently of any particular particle
physics scenario. For a typical size R ~ kpc and compact object masses M ~ 10^7
Msun they remarkably lead to the same particle mass m ~ 10^{-22} eV as the BEC
free-streaming length. However, the phase-space density for the gravitationally
bounded BEC's turns to be more than sixty orders of magnitude smaller than the
galaxy observed values. We conclude that the BEC's and the axion cannot be the
DM particle. However, an axion in the mili-eV scale may be a relevant source of
dark energy through the zero point cosmological quantum fluctuations.
