The star formation history inferred from long gamma-ray bursts with high pseudo-redshifts

•The distribution of pseudo-redshifts of Swift GRBs is presented.•A remarkable number of Pop III GRBs could exist in the present Swift GRB sample.•A plausible constraint is provided on the formation rates of Pop III stars.

By employing a simple semi-analytical star formation model where the formation rates of Population (Pop) I/II and III stars can be calculated, respectively, we account for the number distribution of gamma-ray bursts (GRBs) with high pseudo-redshifts that was derived from an empirical luminosity-indictor relationship. It is suggested that a considerable number of Pop III GRBs could exist in the present sample of Swift   GRBs. By further combining the implication for the star formation history from the optical depth of the CMB photons, it is also suggested that only a very small fraction (∼0.6%∼0.6%) of Pop III GRBs could have triggered the Swift BAT. These results could provide an useful basis for estimating future detectability of Pop III stars and their produced transient phenomena.