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Brightness distribution of Gamma-Ray Bursts (GRBs) is studied in detail under the assumption that GRB rate is related to cosmic star formation rate. The two populations of the long- and short-duration bursts in the 4B BATSE catalog are analyzed separately. Taking account of current uncertainties in the observational estimate of star formation rate (SFR), we have tried various models of the cosmic star formation history and we find that star formation history in relatively low redshifts (z = 0--1) is strongly constrained by the GRB data. In the Einstein-de Sitter universe with (h, Ømega₀, Ømega_\) = (0. 5, 1, 0), the SFR increase factor from z = 0 to 1 should be smaller than 3. 7 (95 % C. L. ) if the rate of the long GRBs is proportional to SFR, and than 5. 8 if the long GRBs are produced by binary neutron-star mergers. These values are significantly smaller than the well-known estimate (\ 14) of the increase factor based on the UV luminosity density, but we argue that this observational value is likely overestimated and the real value may be as low as about 4 if we choose different sets of cosmological parameters and/or take account of the evolution of metallicity and dust extinction. The maximum redshift corresponding to the most faint GRBs observed by the BATSE should be larger than 2, and likely in the range of 3--5. The beaming factor required for the neutron-star merger scenario is about a few hundreds.
Tomonori Totani (Wed,) studied this question.