The stellar content of the Hamburg/ESO survey V. The metallicity distribution function of the Galactic halo

We determine the metallicity distribution function (MDF) of the Galactic halo by means of a sample of 1638 metal-poor stars selected from the Hamburg/ESO objective-prism survey (HES). The sample was corrected for minor biases introduced by the strategy for spectroscopic follow-up observations of the metal-poor candidates, namely best and brightest stars first. Comparison of the metallicities [Fe/H] of the stars determined from moderate-resolution (i.e., R similar to 2000) follow-up spectra with results derived from abundance analyses based on high-resolution spectra (i.e., R > 20 000) shows that the [Fe/H] estimates used for the determination of the halo MDF are accurate to within 0.3 dex, once highly C-rich stars are eliminated. We determined the selection function of the HES, which must be taken into account for a proper comparison between the HES MDF with MDFs of other stellar populations or those predicted by models of Galactic chemical evolution. The latter show a reasonable agreement with the overall shape of the HES MDF for [Fe/H] > -3.6, but only a model of Salvadori et al. (2007) with a critical metallicity for low-mass star formation of Z(cr) = 10(-3.4) Z(circle dot) reproduces the sharp drop at [Fe/H] similar to -3.6 present in the HES MDF. Although currently about ten stars at [Fe/H] < -3.6 are known, the evidence for the existence of a tail of the halo MDF extending to [Fe/H] similar to -5.5 is weak from the sample considered in this paper, because it only includes two stars [Fe/H] < -3.6. Therefore, a comparison with theoretical models has to await larger statistically complete and unbiased samples. A comparison of the MDF of Galactic globular clusters and of dSph satellites to the Galaxy shows qualitative agreement with the halo MDF, derived from the HES, once the selection function of the latter is included. However, statistical tests show that the differences between these are still highly significant.