# The Luminosity Function and Formation Rate of A Complete Sample of Long Gamma-Ray Bursts

## TL;DR

This study analyzes the luminosity function and formation rate of long gamma-ray bursts using a complete, redshift-known sample, revealing evolution patterns and rate densities consistent with or exceeding cosmic star formation rates.

## Contribution

First application of a maximum likelihood method to a complete GRB sample with known redshifts, providing new insights into GRB evolution and rate density.

## Key findings

- Strong redshift evolution in luminosity or density is needed to fit observations.
- GRB formation rate rises steeply up to z=2 and declines thereafter.
- Local GRB rate estimated at approximately 1.5 Gpc^{-3} yr^{-1}.

## Abstract

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 $Swift$ 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 $\delta=2.22^{+0.32}_{-0.31}$) or in density ($\delta=1.92^{+0.20}_{-0.21}$) 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 $z\leq2$ and is proportional to $(1+z)^{-1.07^{+0.98}_{-1.12}}$ for $z\geq2$. 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 $z\leq2$, but shows an enhancement compared to the SFR at $z\geq2$.