  (Abridged) Extensions of the standard model of particle physics predict very
light bosons, ranging from about 10^{-5} eV for the QCD axion to 10^{-33} eV
for ultra-light particles, which could be the cold dark matter (CDM) in the
Universe. If so, their phase-space density must be high enough to form a
Bose-Einstein condensate (BEC). The fluid-like nature of BEC-CDM dynamics
differs from that of standard collisionless CDM (sCDM), so observations of
galactic haloes may distinguish them. sCDM has problems with galaxy
observations on small scales, which BEC-CDM may overcome for a large range of
particle mass m and self-interaction strength g. For quantum-coherence on
galactic scales of radius R and mass M, either the de-Broglie wavelength
lambda_deB <~ R, requiring m >~ m_H \cong 10^{-25}(R/100 kpc)^{-1/2}(M/10^{12}
M_solar)^{-1/2} eV, or else lambda_deB << R but self-interaction balances
gravity, requiring m >> m_H and g >> g_H \cong 2 x 10^{-64} (R/100
kpc)(M/10^{12} M_solar)^{-1} eV cm^3. Here we study the largely-neglected
effects of angular momentum. Spin parameters lambda \cong 0.05 are expected
from tidal-torquing by large-scale structure, just as for sCDM. Since lab BECs
develop quantum vortices if rotated rapidly enough, we ask if this angular
momentum is sufficient to form vortices in BEC haloes, affecting their
structure with potentially observable consequences. The minimum angular
momentum for this, L_{QM} = $\hbar M/m$, requires m >= 9.5 m_H for lambda =
0.05, close to the particle mass required to influence structure on galactic
scales. We study the equilibrium of self-gravitating, rotating BEC haloes which
satisfy the Gross-Pitaevskii-Poisson equations, to calculate if and when
vortices are energetically favoured. Vortices form as long as self-interaction
is strong enough, which includes a large part of the range of m and g of
interest for BEC-CDM haloes.
