**Monthly Notices of the Royal Astronomical Society**

# Angular momentum and vortex formation in Bose–Einstein-condensed cold dark matter haloes

E-mail: daller@astro.as.utexas.edu (TR-D); shapiro@astro.as.utexas.edu (PRS)

## ABSTRACT

Various extensions of the standard model of particle physics predict the existence of very light bosons, with masses ranging from about 10^{−5} eV for the QCD axion down to 10^{−33} eV for ultra-light particles. These particles could be responsible for all or part of the cold dark matter (CDM) in the Universe. For such particles to serve as CDM, 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, however, so different signature effects on galactic haloes may allow observations to distinguish them. Standard CDM has problems with galaxy observations on small scales; cuspy central density profiles of haloes and the overabundance of subhaloes seem to conflict with observations of dwarf galaxies. It has been suggested that BEC-CDM can overcome these shortcomings for a large range of particle mass *m* and self-interaction coupling strength *g*. For quantum coherence to influence structure on the scale of galactic haloes of radius *R* and mass *M*, either the de-Broglie wavelength λ_{deB}≲*R*, which requires *m*≳*m*_{H}≅ 10^{−25}(*R*/100 kpc)^{−1/2}(*M*/10^{12} M_{⊙})^{−1/2} eV, or else λ_{deB}≪*R* but gravity is balanced by self-interaction, which requires *m*≫*m*_{H}*and**g*≫*g*_{H}≅ 2 × 10^{−64}(*R*/100 kpc)(*M*/10^{12} M_{⊙})^{−1} eV cm^{3}. Here we study the largely neglected effects of angular momentum on BEC haloes. Dimensionless spin parameters λ≃ 0.05 are expected from tidal-torquing by large-scale structure formation, just as for standard CDM. Since laboratory BECs develop quantum vortices if rotated rapidly enough, we ask whether this amount of angular momentum is sufficient to form vortices in BEC haloes, which would affect their structure with potentially observable consequences. The minimum angular momentum required for a halo to sustain a vortex, *L*_{QM}, corresponds to ℏ per particle, or ℏ*M*/*m*. For λ= 0.05, this requires *m*≥ 9.5*m*_{H}, close enough to the particle mass required to influence structure on galactic scales that BEC haloes may be subject to vortex formation. While this is a necessary condition, it is not sufficient. To determine if and when quantum vortices will form in BEC haloes with a given λ-value, we study the equilibrium of self-gravitating, rotating, virialized BEC haloes which satisfy the Gross–Pitaevskii–Poisson equations, and calculate under what conditions vortices are energetically favoured, in two limits: either just enough angular momentum for one vortex or a significant excess of angular momentum. For λ= 0.05, vortex formation is energetically favoured for *L*/*L*_{QM}≥ 1 as long as *both**m*/*m*_{H}≥ 9.5 *and**g*/*g*_{H}≥ 68.0. Hence, vortices are expected for a wide range of BEC parameters. However, vortices cannot form for vanishing self-interaction (i.e. when λ_{deB}≲*R*), and a range of particle parameters also remain even for BEC haloes supported by self-interaction, for which vortices will *not* form. Such BEC haloes can be modelled by compressible, (*n*= 1)-polytropic, irrotational Riemann-S ellipsoids.