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tokens: high-redshift probe of the first stars. Figure 2. Confronting structure formation theory with observations. Shown is the critical (minimum) DM halo mass required for the runaway collapse of primordial gas versus redshift for different cosmological models (see Fig. 1 ). The three coloured lines represent the critical halo properties for CDM ( blue ) as well as FDM models with ( orange ) and ( green ). The symbols indicate 2σ, 3σ, and 4σ peaks, respectively. The magenta region summarizes estimates for the DM halo mass, hosting UFD galaxies in the Local Group. Complementary to this, the light-grey region delineates the redshift range inferred from the observed EDGES spectral trough in the 21-cm absorption profile (Bowman et al. 2018 ). The dark-grey region shows the epoch of reionization. As is evident, the combined empirical constraints disfavour FDM models with the lightest axion masses. 3.1 Local Group dwarf galaxies The first constraint is the mass limit for the first star-forming haloes, M crit , such that at a given redshift runaway cooling can only occur in sufficiently massive hosts. Observations of UFD galaxies provide non-trivial tests for any model of structure formation, as the model must allow star formation in low-mass host haloes, sufficiently early in cosmic history. Possibly, masses of the UFD progenitors may be as small as at a redshift of z ≈ 10 (Bland-Hawthorn, Sutherland Webster 2015 ). However, observations of individual UFDs in the present-day Universe give relatively uncertain virial masses of the order of (e.g. Wolf et al. 2010 ; Simon et al. 2011 ; Boylan-Kolchin, Bullock Kaplinghat 2012 ), and a recent statistical analysis of local UFDs provides a conservative upper limit of the minimum UFD-hosting halo mass of (Jethwa, Erkal Belokurov 2018 ). We illustrate the plausible range of UFD halo masses in Fig. 2 . Structure formation models represented by curves that lie beneath the upper limit of this region are consistent with the Local Group constraints. In the ΛCDM cosmology, M crit is always below the region constrained by the UFD observations, regardless of overdensity νσ. This is in accordance with the standard picture of hierarchical structure formation. For the FDM model with the lighter axion, , on the other hand, M crit lies above the allowed UFD region for 2σ peaks. In this case, the most frequently occurring haloes would form with masses larger than what is inferred for the UFD hosts. The resulting tension implies that existing UFD observations disfavour the lighter-axion FDM model. Conversely, this tension does not exist in the model with heavier-axion FDM (). Here, values of M crit for reasonable, not too unlikely, σ peaks lie within or below the allowed UFD region. Though it remains unclear whether UFDs alone can rule out FDM entirely, further restrictions for the allowed models are anticipated through observations of lower-mass UFDs with upcoming large-scale surveys, carried out, e.g., with the Large Synoptic