Angular Momentum and the Absence of Vortices in the Cores of Fuzzy Dark
  Matter Haloes
  Scalar Field Dark Matter (SFDM), comprised of ultralight ($\gtrsim 10^{-22}$
eV) bosons, is distinguished from massive ($\gtrsim$ GeV), collisionless Cold
Dark Matter (CDM) by its novel structure-formation dynamics as Bose-Einstein
condensate (BEC) and quantum superfluid with wave-like properties, described by
the Gross-Pitaevski and Poisson (GPP) equations. In the free-field (fuzzy)
limit of SFDM (FDM), structure is inhibited below the de Broglie wavelength
$\lambda_{\text{deB}}$, but resembles CDM on larger scales. Virialized haloes
have solitonic cores of radius $\sim \lambda_{\text{deB}}$ that follow the
ground-state attractor solution of GPP, surrounded by CDM-like envelopes. As
superfluid, SFDM is irrotational (vorticity-free) but can be unstable to vortex
formation. We previously showed this can happen in halo cores, from angular
momentum arising during structure formation, when repulsive self-interaction
(SI) is present to support them out to a second length scale
$\lambda_{\text{SI}}$ with $\lambda_{\text{SI}} > \lambda_{\text{deB}}$ (the
Thomas-Fermi regime), but only if SI is strong enough. This suggested FDM cores
(without SI) would not form vortices. FDM simulations later found vortices, but
only outside halo cores, consistent with our previous suggestion based upon
TF-regime analysis. We extend that analysis now to FDM, to show explicitly that
vortices should not arise in solitonic cores from angular momentum, modelling
them as either Gaussian spheres or compressible, ($n = 2$)-polytropic,
irrotational Riemann-S ellipsoids. We find that, for typical halo spin
parameters, angular momentum per particle is below $\hbar$, the minimum
required even for one singly-quantized vortex in the centre. Even for larger
angular momentum, however, vortex formation is not energetically favoured.
