The rate and luminosity function of long gamma ray bursts

¹ Universitá degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy
e-mail: alessio.pescalli@brera.inaf.it
² INAF–Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate, Italy
³ INAF–IASF Milano, via E. Bassini 15, 20133 Milano, Italy
⁴ GEPI, Observatoire de Paris, PSL Research University, CNRS, Université Paris Diderot, Sorbonne Paris Cité, Place Jules Janssen, 92195 Meudon, France
⁵ Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
⁶ AIM, UMR 7158 CEA/DSM-CNRS-Université Paris Diderot, Irfu/Service d’Astrophysique, Saclay, 91191 Gif-sur-Yvette Cedex, France

Received: 16 June 2015
Accepted: 13 November 2015

Abstract

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 (~82%). We parametrise the redshift evolution of the GRB luminosity as L = L₀(1 + z)^k and we derive k = 2.5, consistently with recent estimates. The de-evolved luminosity function φ(L₀) of GRBs can be 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,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 ~ 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.