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Scalar Field Dark Matter (SFDM) comprised of ultralight ($\gtrsim 10^{-22}$eV) bosons is an alternative to standard, collisionless Cold Dark Matter (CDM)that is CDM-like on large scales but inhibits small-scale structure formation.As a Bose-Einstein condensate, its free-field ("fuzzy") limit (FDM) suppressesstructure below the de Broglie wavelength, $\lambda_\text{deB}$, creatingvirialized haloes with central cores of radius $\sim\lambda_\text{deB}$,surrounded by CDM-like envelopes, and a halo mass function (HMF) with a sharpcut-off on small scales. With a strong enough repulsive self-interaction (SI),structure is inhibited, instead, below the Thomas-Fermi (TF) radius,$R_\text{TF}$ (the size of an SI-pressure-supported ($n=1$)-polytrope), when$R_\text{TF} > \lambda_\text{deB}$. Previously, we developed tools to describeSFDM dynamics on scales above $\lambda_\text{deB}$ and showed that SFDM-TFhaloes formed by Jeans-unstable collapse from non-cosmological initialconditions have $R_\text{TF}$-sized cores, surrounded by CDM-like envelopes.Revisiting SFDM-TF in the cosmological context, we simulate halo formation bycosmological infall and collapse, and derive its transfer function from linearperturbation theory to produce cosmological initial conditions and predictstatistical measures of structure formation, such as the HMF. Since FDM andSFDM-TF transfer functions both have small-scale cut-offs, we can align them tolet observational constraints on FDM proxy for SFDM-TF, finding FDM withparticle masses $1 \lesssim m/(10^{-22} \text{ eV}/c^2) \lesssim 30$corresponds to SFDM-TF with $10 \gtrsim R_\text{TF}/(1 \text{ pc}) \gtrsim 1$,favoring sub-galactic (sub-kpc) core-size. The SFDM-TF HMF cuts off gradually,however, leaving more small-mass haloes: its Jeans mass shrinks so fast, scalesfiltered early can still recover and grow!

Cosmological structure formation in scalar field dark matter with repulsive self-interaction: The Incredible Shrinking Jeans Mass