A Comprehensive Study of Luminosity Functions and Event Rate Densities of Long Gamma-Ray Bursts with Non-Parametric Method

The current event rate estimates of long gamma-ray bursts based on distinct methods or samples especially at lower redshift are largely debated, which motivates us to re-study the dependence of luminosity function and event rates for different burst samples on the criteria of sample selection and threshold effect in this letter. To ensure the sample completeness as possible, we have chosen two samples including 88 and 118 long bright bursts with known redshift and peak flux over 2.6 ph cm$^{-2}$ s$^{-1}$. It is found that the evolution of luminosity with redshift can be expressed by $Lpropto(1+z)^k$ with a diverse $k$ relied more on the sample selection. Interestingly, the cumulative distributions of either non-evolving luminosities or redshifts are found to be also determined by the sample selection rather the instrumental sensitivity. Nevertheless, the non-evolving luminosities of our samples are similarly distributed with a comparable break luminosity of $L_0sim10^{51}$ erg s$^{-1}$. Importantly, we verify with a K-S test that three cases of event rates for the two burst samples evolve with redshift similarly except a small discrepancy due to sampling differences at low-redshift of $z<1$, in which all event rates show an excess of gaussian profile instead of monotonous decline. Most importantly, it is found that the low-redshift burst event rates violate the star formation rates, while both of them are good in agreement with each other in the higher-redshift regions as many authors discovered previously. Consequently, we predict that two types of long gamma-ray bursts should be expected on the basis of whether they match the star formation or not.