  For ultra-light scalar particles like axions, dark matter can form a state of
the Bose-Einstein condensate (BEC) with a coherent classical wave whose
wavelength is of order galactic scales. In the context of an oscillating scalar
field with mass $m$, this BEC description amounts to integrating out the field
oscillations over the Hubble time scale $H^{-1}$ in the regime $m \gg H$. We
provide a gauge-invariant general relativistic framework for studying
cosmological perturbations in the presence of a self-interacting BEC associated
with a complex scalar field. In particular, we explicitly show the difference
of BECs from perfect fluids by taking into account cold dark matter, baryons,
and radiation as a Schutz-Sorkin description of perfect fluids. We also
scrutinize the accuracy of commonly used Newtonian treatment based on a
quasi-static approximation for perturbations deep inside the Hubble radius. For
a scalar field which starts to oscillate after matter-radiation equality, we
show that, after the BEC formation, a negative self-coupling hardly leads to a
Laplacian instability of the BEC density contrast. This is attributed to the
fact that the Laplacian instability does not overwhelm the gravitational
instability for self-interactions within the validity of the nonrelativistic
BEC description. Our analysis does not accommodate the regime of parametric
resonance which can potentially occur for a large field alignment during the
transient epoch prior to the BEC formation.
