Limits on Population III star formation with the most iron- poor stars

We study the impact of star-forming minihaloes, and the initial mass function (IMF) of Popu-lation III (Pop III) stars, on the Galactic halo metallicity distribution function (MDF) and on the properties of C-enhanced and C-normal stars at [Fe/H] < -3. For our investigation we use a data-constrained merger tree model for the MilkyWay formation, which has been improved to self-consistently describe the physical processes regulating star formation in minihaloes, including the poor sampling of the Pop III IMF. We find that only when star-forming minihaloes are included the low-Fe tail of the MDF is correctly reproduced, showing a plateau that is built up by C-enhanced metal-poor stars imprinted by primordial faint supernovae. The incomplete sampling of the Pop III IMF in inefficiently star-forming minihaloes (< 10(-3)M(circle dot) yr(-1)) strongly limits the formation of pair-instability supernovae (PISNe), with progenitor masses m(PopIII) = [140-260] M-circle dot, even when a flat Pop III IMF is assumed. Second-generation stars formed in environments polluted at > 50 per cent level by PISNe are thus extremely rare, corresponding to approximate to 0.25 per cent of the total stellar population at [Fe/H] < -2, which is consistent with recent observations. The low-Fe tail of the MDF strongly depends on the Pop III IMF shape and mass range. Given the current statistics, we find that a flat Pop III IMF model with mPopIII = [10-300] M-circle dot is disfavoured by observations. We present testable predictions for Pop III stars extending down to lower masses, with mPopIII = [0.1-300] M-circle dot