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Bose-Einstein-condensed dark matter (BEC-DM), also called scalar field dark matter (SFDM), has become a popular alternative to the standard, collisionless cold dark matter (CDM) model, due to its long-held potential to resolve the small-scale crisis of CDM. Halos made of BEC-DM have been modelled using the Gross-Pitaevskii (GP) equation coupled to the Poisson equation; the so-called GPP equations of motion. These equations are based on fundamental microphysical conditions that need to be fulfilled in order for the equations to be valid in the first place, related to the diluteness of the DM gas and the nature of the particle scattering model. We use these conditions in order to derive the implications for the BEC-DM parameters, the 2-particle self-interaction coupling strength $g$ and the particle mass $m$. We compare the derived bounds with the constraint that results from the assumption of virial equilibrium of the central cores of halos, deriving a relationship that connects $g$ and $m$. We find that the GPP conditions are greatly fulfilled, for BEC-DM particle masses of interest, if such models also obey the virial condition that turns out to be the strongest constraint. We also derive the implications for the elastic scattering cross section (per particle mass) in BEC-DM halos, based on the scattering model of GPP, and find a huge range of possible values, depending on the self-interaction regime. We put our results into context to recent literature which predicts sub-kpc core size in BEC-DM halos., Comment: 24 pages; 3 figures; revised version which includes more explanations and references and an expanded Figure 1

(abridged) We present simulations of halo formation and evolution in scalar field dark matter (SFDM) cosmologies in the Thomas-Fermi regime, aka ``SFDM-TF", where a strong repulsive 2-particle self-interaction (SI) is included, being a valuable alternative to CDM, with the potential to resolve its ``cusp-core" problem. In general, SFDM behaves like a quantum fluid. Previous literature has presented two fluid approximations for SFDM-TF, as well as simulations of halo formation. These results confirmed earlier expectations and are generally in mutual agreement, but discrepancies were also reported. Therefore, we perform dedicated 3D cosmological simulations for the SFDM-TF model, applying both fluid approximations, as well as for CDM. Our results are very well in accordance with previous works and extend upon them, in that we can explain the reported discrepancies as a result of different simulation setups. We find some interesting details: The evolution of both SFDM-TF and CDM halos follows a 2-stage process. In the early stage, the density profile in the center becomes close to a $(n=1.5)$-polytropic core, dominated by an "effective" velocity-dispersion pressure $P_{\sigma}$ which is common to both dark matter models. Consecutively, for CDM halos, the core transitions into a central cusp. In SFDM-TF halos, the additional pressure $P_\text{SI}$ due to SI determines the second stage of the evolution, where the central region follows closely a $(n=1)$-polytropic core, embedded in a nearly isothermal envelope, i.e. the outskirts are similar to CDM. We also encounter a new effect, namely a late-time expansion of both polytropic core plus envelope, because the size of the almost isothermal halo envelope is affected by the expansion of the background universe. So, an initial primordial core of $\sim 100$ pc can evolve into a larger core of $\gtrsim 1$ kpc, even without feedback from baryons., Comment: submitted to Phys.Rev.D; 26 pages, 16 figures