  We analyze the rotation curves that correspond to a Bose--Einstein Condensate
(BEC) type halo surrounding a Schwarzschild--type black hole to confront
predictions of the model upon observations of galaxy rotation curves. We model
the halo as a Bose--Einstein condensate in terms of a massive scalar field that
satisfies a Klein--Gordon equation with a self--interaction term. We also
assume that the bosonic cloud is not self--gravitating. To model the halo, we
apply a simple form of the Thomas--Fermi approximation that allows us to
extract relevant results with a simple and concise procedure. Using galaxy data
from a subsample of SPARC data base, we find the best fits of the BEC model by
using the Thomas--Fermi approximation and perform a Bayesian statistics
analysis to compare the obtained BEC's scenarios with the Navarro--Frenk--White
(NFW) model as pivot model. We find that in the centre of galaxies we must have
a supermassive compact central object, i.e., supermassive black hole, in the
range of $\log_{10} M/M_\odot = 11.08 \pm 0.43$ which condensate a boson cloud
with average particle mass $M_\Phi = (3.47 \pm 1.43 )\times10^{-23}$ eV and a
self--interaction coupling constant $\log_{10} (\lambda \; [{\rm pc}^{-1}]) =
-91.09 \pm 0.74 $, i.e., the system behaves as a weakly interacting BEC. We
compare the BEC model with NFW concluding that in general the BEC model using
the Thomas--Fermi approximation is strong enough compared with the NFW
fittings. Moreover, we show that BECs still well--fit the galaxy rotation
curves and, more importantly, could lead to an understanding of the dark matter
nature from first principles.
