We study the luminosity function and formation rate of long gamma-ray bursts (GRBs) by using a maximum likelihood method. This is the first time this method is applied to a well-defined sample of GRBs that is complete in redshift. The sample is composed of 99 bursts detected by the Swif t satellite, 81 of them with measured redshift and luminosity for a completeness level of 82%. We confirm that a strong redshift evolution in luminosity (with an evolution index of δ = 2.22 +0.32 −0.31 ) or in density (δ = 1.92 +0.20 −0.21 ) is needed in order to reproduce the observations well. But since the predicted redshift and luminosity distributions in the two scenarios are very similar, it is difficult to distinguish between these two kinds of evolutions only on the basis of the current sample. Furthermore, we also consider an empirical density case in which the GRB rate density is directly described as a broken power-law function and the luminosity function is taken to be non-evolving. In this case, we find that the GRB formation rate rises like (1 + z) 3.85 +0.48 −0.45 for z < ∼ 2 and is proportional to (1 + z) −1.07 +0.98 −1.12 for z > ∼ 2. The local GRB rate is 1.49 +0.63 −0.64 Gpc −3 yr −1 . The GRB rate may be consistent with the cosmic star formation rate (SFR) at z < ∼ 2, but shows an enhancement compared to the SFR at z > ∼ 2.

In this work, we update and enlarge the long gamma-ray burst (GRB) sample detected by the Swift satellite. Given the incomplete sampling of the faint bursts and the low completeness in redshift measurement, we carefully select a subsample of bright Swift bursts to revisit the GRB luminosity function (LF) and redshift distribution by taking into account the probability of redshift measurement. Here we also explore two general expressions for the GRB LF, i.e. a broken power-law LF and a triple power-law LF. Our results suggest that a strong redshift evolution in luminosity (with an evolution index of $\delta =1.92^{+0.25}_{-0.37}$) or in density ($\delta =1.26^{+0.33}_{-0.34}$) is required in order to well account for the observations, independent of the assumed expression of the GRB LF. However, in a one-on-one comparison using the Akaike information criterion, the best-fitting evolution model involving the triple power-law LF is statistically preferred over the best-fitting one involving the broken power-law LF with a relative probability of ∼94.3 per cent versus ∼5.7 per cent. Extrapolating our fitting results to the flux limit of the whole Swift sample, and considering the trigger probability of Swift/Burst Alert Telescope in detail, we find that the expectations from our evolution models provide a good representation of the observed distributions of the whole sample without the need for any adjustment of the model free parameters. This further confirms the reliability of our analysis results.

Long gamma-ray bursts (GRBs) have been discussed as a potential tool to probe the cosmic star formation rate (SFR) for a long time. Some studies found an enhancement in the GRB rate relative to the galaxy-inferred SFR at high redshifts, which indicates that GRBs may not be good tracers of star formation. However, in these studies, the GRB rate measured at any redshift is an average over all galaxies at that epoch. A deep understanding of the connection between GRB production and environment also needs to characterize the population of GRB host galaxies directly. Based on a complete sample of GRB hosts, we constrain the stellar-mass function (SMF) of GRB hosts, and examine redshift evolution in the GRB host population. Our results confirm that a strong redshift evolution in energy (with an evolution index of δ = 2.47 − 0.89 + 0.73 ) or in density ( δ = 1.82 − 0.59 + 0.22 ) is needed in order to account for the observations. The GRB host SMF can be well described by the Schechter function with a power-law index ξ ≈ −1.10 and a break mass M b,0 ≈ 4.9 × 1010 M ⊙, independent of the assumed evolutionary effects. This is the first formulation of the GRB host SMF. The observed discrepancy between the GRB rate and the galaxy-inferred SFR may also be explained by an evolving SMF.