The question how much star formation is occurring at low metallicity throughout the cosmic history appears crucial for the discussion of the origin of various energetic transients, and possibly - double black hole mergers. We revisit the observation-based distribution of birth metallicities of stars (f$_{rm SFR}$(Z,z)), focusing on several factors that strongly affect its low metallicity part: (i) the method used to describe the metallicity distribution of galaxies (redshift-dependent mass metallicity relation - MZR, or redshift-invariant fundamental metallicity relation - FMR), (ii) the contribution of starburst galaxies and (iii) the slope of the MZR. We empirically construct the FMR based on the low-redshift scaling relations, which allows us to capture the systematic differences in the relation caused by the choice of metallicity and star formation rate (SFR) determination techniques and discuss the related f$_{rm SFR}$(Z,z) uncertainty. We indicate factors that dominate the f$_{rm SFR}$(Z,z) uncertainty in different metallicity and redshift regimes. The low metallicity part of the distribution is poorly constrained even at low redshifts (even a factor of $sim$200 difference between the model variations) The non-evolving FMR implies a much shallower metallicity evolution than the extrapolated MZR, however, its effect on the low metallicity part of the f$_{rm SFR}$(Z,z) is counterbalanced by the contribution of starbursts (assuming that they follow the FMR). A non-negligible fraction of starbursts in our model may be necessary to satisfy the recent high-redshift SFR density constraints.
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