  A complex scalar field has recently been suggested to bind galaxies and
flatten the rotation curves of spirals. Its cosmological behavior is thoroughly
investigated here. Such a field is shown to be a potential candidate for the
cosmological dark matter that fills up a fraction Omega_cdm = 0.3 of the
Universe. However, problems arise when the limits from galactic dynamics and
some cosmological constraints are taken simultaneously into account. A free
complex field, associated to a very small mass m = 10^{-23} eV, has a correct
cosmological behavior in the early Universe, but behaves today mostly as a real
axion, with a problematic value of its conserved quantum number. On the other
hand, an interacting field with quartic coupling lambda = 0.1 has a more
realistic mass m = 1 eV and carries a quantum number close to the photon number
density. Unlike a free field, it would be spinning today in the complex plane -
like the so-called ``spintessence''. Unfortunately, the cosmological evolution
of such field in the early Universe is hardly compatible with constraints from
nucleosynthesis and structure formation.
