Bose-Einstein Condensate Dark Matter (BECDM; also known as Fuzzy Dark Matter) is motivated by fundamental physics and has recently received significant attention as a serious alternative to the established Cold Dark Matter (CDM) model. We perform cosmological simulations of BECDM gravitationally coupled to baryons and investigate structure formation at high redshifts ($z gtrsim 5$) for a boson mass $m=2.5cdot 10^{-22}~{rm eV}$, exploring the dynamical effects of its wavelike nature on the cosmic web and the formation of first galaxies. Our BECDM simulations are directly compared to CDM as well as to simulations where the dynamical quantum potential is ignored and only the initial suppression of the power spectrum is considered -- a Warm Dark Matter-like (WDM) model often used as a proxy for BECDM. Our simulations confirm that WDM is a good approximation to BECDM on large cosmological scales even in the presence of the baryonic feedback. Similarities also exist on small scales, with primordial star formation happening both in isolated haloes and continuously along cosmic filaments; the latter effect is not present in CDM. Global star formation and metal enrichment in these first galaxies are delayed in BECDM/WDM compared to the CDM case: in BECDM/WDM first stars form at $zsim 13$/$13.5$ while in CDM star formation starts at $zsim 35$. The signature of BECDM interference, not present in WDM, is seen in the evolved dark matter power spectrum: although the small scale structure is initially suppressed, power on kpc scales is added at lower redshifts. Our simulations lay the groundwork for realistic simulations of galaxy formation in BECDM.
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