Selection effects in gamma-ray burst correlations: Consequences on the ratio between gamma-ray burst and star formation rates

Gamma-ray bursts (GRBs) visible up to very high redshift have become attractive targets as potential new distance indicators. It is still not clear whether the relations proposed so far originate from an unknown GRB physics or result from selection effects. We investigate this issue in the case of the LX - (hereafter LT) correlation between the X-ray luminosity LX (Ta ) at the end of the plateau phase, Ta , and the rest-frame time . We devise a general method to build mock data sets starting from a GRB world model and taking into account selection effects on both time and luminosity. This method shows how not knowing the efficiency function could influence the evaluation of the intrinsic slope of any correlation and the GRB density rate. We investigate biases (small offsets in slope or normalization) that would occur in the LT relation as a result of truncations, possibly present in the intrinsic distributions of LX and . We compare these results with the ones in Dainotti et al. showing that in both cases the intrinsic slope of the LT correlation is -1.0. This method is general and therefore relevant for investigating whether or not any other GRB correlation is generated by the biases themselves. Moreover, because the farthest GRBs and star-forming galaxies probe the reionization epoch, we evaluate the redshift-dependent ratio Ψ(z) = (1 + z)αof the GRB rate to the star formation rate. We found a modest evolution -0.2 ≤ α ≤ 0.5 consistent with a Swift GRB afterglow plateau in the redshift range 0.99 < z < 9.4.