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

We study the impact of star-forming mini-haloes, and the Initial Mass Function (IMF) of Population 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 Milky Way formation, which has been improved to self-consistently describe the physical processes regulating star-formation in mini-haloes, including the poor sampling of the Pop III IMF. We find that only when star-forming mini-haloes are included the low-Fe tail of the MDF is correctly reproduced, showing a plateau that is built up by C-enhanced metal-poor (CEMP) stars imprinted by primordial faint supernovae. The incomplete sampling of the Pop III IMF in inefficiently star-forming mini-haloes (< $10^{-3}$ $M_\odot$/yr) strongly limits the formation of Pair Instability Supernovae (PISNe), with progenitor masses $m_{\rm popIII}$=[140-260] $M_\odot$, even when a flat Pop III IMF is assumed. Second-generation stars formed in environments polluted at >50% level by PISNe are thus extremely rare, corresponding to $\approx$ 0.25% 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 $m_{\rm popIII}$=[10-300] $M_\odot$ is disfavoured by observations. We present testable predictions for Pop III stars extending down to lower masses, with $m_{\rm popIII}$=[0.1-300] $M_\odot$.