Core-Envelope Haloes in Scalar Field Dark Matter with Repulsive Self-Interaction: Fluid Dynamics Beyond the de Broglie Wavelength

Scalar Field Dark Matter (SFDM) comprised of ultralight bosons has attracted great interest as an alternative to standard, collisionless Cold Dark Matter (CDM) because of its novel structure-formation dynamics, described by the coupled Schr\"odinger-Poisson equations. In the free-field ("fuzzy") limit of SFDM (FDM), structure is inhibited below the de Broglie wavelength, but resembles CDM on larger scales. Virialized haloes have "solitonic" cores of radius $\sim\lambda_\text{deB}$, surrounded by CDM-like envelopes. When a strong enough repulsive self-interaction (SI) is also present, structure can be inhibited below a second length scale, $\lambda_\text{SI}$, with $\lambda_\text{SI}> \lambda_\text{deB}$ -- called the Thomas-Fermi (TF) regime. FDM dynamics differs from CDM because of quantum pressure, and SFDM-TF differs further by adding SI pressure. In the small-$\lambda_\text{deB}$ limit, however, we can model all three by fluid conservation equations for a compressible, $\gamma=5/3$ ideal gas, with ideal gas pressure sourced by internal velocity dispersion and, for the TF regime, an added SI pressure, $P_\text{SI}\propto \rho^2$. We use these fluid equations to simulate halo formation from gravitational collapse in 1D, spherical symmetry, demonstrating for the first time that SFDM-TF haloes form with cores the size of $R_\text{TF}$, the radius of an SI-pressure-supported $(n=1)$-polytrope, surrounded by CDM-like envelopes. In comparison with rotation curves of dwarf galaxies in the local Universe, SFDM-TF haloes pass the ["too-big-to-fail" + "cusp-core"]-test if $R_\text{TF}\gtrsim 1$ kpc.