Long-duration Gamma Ray Bursts (GRBs) are linked to the collapse of massive stars and their hosts are exclusively identified as active, star forming galaxies. Four long GRBs observed at high spectral resolution at redshift 1.5 <z < 4 allowed the determination of the elemental abundances for a set of different chemical elements. In this paper, for the first time, by means of detailed chemical evolution models taking into account also dust production, we attempt to constrain the star formation history of the host galaxies of these GRBs from the study of the chemical abundances measured in their ISM. We are also able to provide constraints on the age and on the dust content of GRB hosts. Our results support the hypothesis that long duration GRBs occur preferentially in low metallicity, star forming galaxies. We compare the specific star formation rate, namely the star formation rate per unit stellar mass, predicted for the hosts of these GRBs with observational values for GRB hosts distributed across a large redshift range. Our models predict a decrease of the specific star formation rate (SSFR) with redshift, consistent with the observed decrease of the comoving cosmic SFR density between z ~2 and z=0. On the other hand, observed GRB hosts seems to follow an opposite trend in the SSFR vs redshift plot, with an increase of the SSFR with decreasing redshift. Finally, we compare the SSFR of GRB050730 host with values derived for a sample of Quasar damped Lyman alpha systems. Our results indicate that the abundance pattern and the specific star formation rates of the host galaxies of these GRBs are basically compatible with the ones determined in Quasar damped Lyman alpha systems, suggesting similar chemical evolution paths.
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