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 } \sim 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 \sim 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 \sim 0.1 has a more realistic mass m \sim 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 .