  In the last time the cold dark matter (CDM) model has suggested more and more
that it is not able to describe all the properties of nearby galaxies that can
be observed in great detail as well as that it has some problems in the
mechanism by which matter is more rapidly gathered into large-scale structure
such as galaxies and clusters of galaxies. In this work we revisit an
alternative model, the scalar field dark matter (SFDM) model, which proposes
that the galactic haloes form by condensation of a scalar field (SF) very early
in the Universe, i.e., in this model the haloes of galaxies are astronomical
Bose-Einstein Condensate drops of SF. On the other hand, large-scale structures
like clusters or superclusters of galaxies form similar to the $\Lambda$CDM
model, by hierarchy, thus all the predictions of the $\Lambda$CDM model at
cosmological scales are reproduced by SFDM. This model predicts that all galaxy
haloes must be very similar and exist for higher redshifts than in the
$\Lambda$CDM model. In the first part of this review we revisit the
cosmological evolution of SFDM model with a scalar potential
$m^2\Phi^2/2+\lambda\Phi^4/4$ with two different frameworks: the field and
fluid approach. The scalar fluctuations have an oscillating growing mode and
therefore, this kind of dark matter could lead to the early formation of
gravitational structures in the Universe. In the last part, we study the core
central density profiles of BEC dark matter haloes and fit high-resolution
rotation curves of low surface brightness galaxies. The mean value of the
logarithmic inner density slopes is $\alpha = - $0.27 $\pm$ 0.18 and we show
that the recent observation of the constant dark matter central surface density
can be reproduced. We conclude that in light of the difficulties that the
$\Lambda$CDM model is currently facing the SFDM model can be a worthy
alternative to keep exploring further.
