We investigate the metallicity distribution function ( MDF ) in the Galactic halo and the relative fraction of Carbon-normal and Carbon-rich stars .
To this aim , we use an improved version of the semi-analytical code GA laxy ME rger T ree and E volution ( gamete ) , that reconstructs the hierarchical merger tree of the MW , following the star formation history and the metal and dust evolution in individual progenitors .
The predicted scaling relations between the dust , metal and gas masses for MW progenitors show a good agreement with observational data of local galaxies and of Gamma Ray Burst ( GRB ) host galaxies at 0.1 < z < 6.3 .
Comparing the simulated and the observed MDF , we find that in order to predict the formation of hyper-iron-poor stars at [ Fe / H ] < -4 , faint SN explosions have to dominate the metal yields produced by Pop III stars , disfavoring a Pop III IMF that extends to stellar masses > 140 M _ { \odot } , into the Pair-Instability SN progenitor mass range .
The relative contribution of C-normal and C-enhanced stars to the MDF and its dependence on [ Fe/H ] points to a scenario where the Pop III/II transition is driven by dust-cooling and the first low-mass stars form when the dust-to-gas ratio in their parent clouds exceeds a critical value of { \cal D } _ { crit } = 4.4 \times 10 ^ { -9 } .
Other transition criteria do not predict any C-normal stars below [ Fe/H ] < -4 , at odds with observations .