The rate and luminosity function of long gamma ray bursts

We derive, adopting a direct method, the luminosity function and the formation rate of long Gamma Ray Bursts through a complete, flux-limited, sample of Swift bursts which has a high level of completeness in redshift Z (similar to 82%). We parametrise the redshift evolution of the GRB luminosity as L = L-0(1 + z)(k) and we derive k = 2.5, consistently with recent estimates. The de-evolved luminosity function phi(L-0) of GRBs can he represented by a broken power law with slopes a = -1.32 +/- 0.21 and b = -1.84 +/- 0.24 below and above, respectively, a break luminosity L-0,L-b = 10(51.45 +/- 0.15) erg/s. Under the hypothesis of luminosity evolution we find that the GRB formation rate increases with redshift up to z similar to 2, where it peaks, and then decreases in agreement with the shape of the cosmic star formation rate. We test the direct method through numerical simulations and we show that if it is applied to incomplete (both in redshift and/or flux) GRB samples it can misleadingly result in an excess of the GRB formation rate at low redshifts.