Angular momentum and the absence of vortices in the cores of fuzzy dark matter haloes

Scalar field dark matter (SFDM), comprised of ultralight (≳ 10^-22 eV) bosons, is distinguished from massive (≳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-Pitaevskii and Poisson (GPP) equations. In the free-field ('fuzzy') limit of SFDM (FDM), structure is inhibited below the de Broglie wavelength λ_deB, but resembles CDM on larger scales. Virialized haloes have 'solitonic' cores of radius ~λ_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 λ_SI with λ_SI > λ_deB (the Thomas-Fermi regime), but only if SI is strong enough. This suggested FDM cores ($ {\rm 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 ( n = 2)-polytropic, irrotational Riemann-S ellipsoids. We find that, for typical halo spin parameters, angular momentum per particle is below ℏ, the minimum required even for one singly-quantized vortex in the centre. Even for higher angular momentum, however, vortex formation is not energetically favoured.