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) suppresses structure below the de Broglie wavelength, $lambda_text{deB}$, creating virialized haloes with central cores of radius $simlambda_text{deB}$, surrounded by CDM-like envelopes, and a halo mass function (HMF) with a sharp cut-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 describe SFDM dynamics on scales above $lambda_text{deB}$ and showed that SFDM-TF haloes formed by Jeans-unstable collapse from non-cosmological initial conditions have $R_text{TF}$-sized cores, surrounded by CDM-like envelopes. Revisiting SFDM-TF in the cosmological context, we simulate halo formation by cosmological infall and collapse, and derive its transfer function from linear perturbation theory to produce cosmological initial conditions and predict statistical measures of structure formation, such as the HMF. Since FDM and SFDM-TF transfer functions both have small-scale cut-offs, we can align them to let observational constraints on FDM proxy for SFDM-TF, finding FDM with particle 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, scales filtered early can still recover and grow!
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