The Evolution Of The Formation Rate And Luminosity Function Of Long-duration Gamma-ray Bursts

Gamma-ray bursts(GRBs)are brief flashes of gamma-rays occurring at cosmological distances.Lasting from a few milliseconds to several minutes,GRBs typically have isotropic-equivalent luminosity Liso～1046-55 erg s-1.According to their durations(T90),GRBs are generally classified into two categories.For bursts with duration T90<2s,we call them short GRBs(sGRBs)which are believed to originate from the merger of compact binary systems.And the otherwise are long GRBs(LGRBs)which are associated with the death of massive stars.LGRBs usually shine hundreds of times brighter than ordinary supernovae.Thanks to their high brightness,LGRBs can be detected at very far distances(up to z～9.23 currently).In chapter 1,we briefly introduce the observational characteristics and theoretical models of GRBs.This thesis mainly focuses on the links between the formation rates of LGRBs and stars.Since the lifespan of massive stars are very short,their formation and death can be considered to be on the same time scale,which means that LGRBs should be a useful tool to trace the star formation rate(SFR).And considering they are so bright relative to ordinary optical systems that can be detected at very far distances,LGRBs therefore provide an excellent method to measure the high-z SFR in principle.However,some works find that the LGRB rate does not strictly follow the SFR,but shows an enhancement at high redshifts.We intently investigate this open question,and the relevant progresses in recent years are introduced in chapter 2 and a part of chapter 5.In chapter 3,we estimate the luminosity and redshift distributions of long GRBs based on a complete Swift sample which is constructed for reducing the bias caused by instrument selection effects.Previous works have used non-parametric methods(e.g.the Lynden-Bell method)to estimate the expected distributions of this complete sample.However,this method requires evenly distributed data which is difficult to achieve for GRBs as their sample size is too small.Then incalculable errors may appear in the results.Here we use the maximum likelihood method,for the first time,to estimate the LGRB redshift and luminosity distributions of this sample with models such as simple(no evolution)star formation model,density evolution model,luminosity evolution model,and empirical model.Our results are similar to previous works,the empirical model and two evolutionary models can reproduce the observed redshift distribution very well,while the no evolution model significantly underestimates the observation at high redshifts.We also compare the empirical rate with the SFR,and find that the LGRB rate is indeed much higher than the SFR at high redshifts.Although complete samples can reduce the systematic error of redshift measurements and improve the completeness of redshift and trigger in samples,it is hard to guarantee that such complete samples reflect intrinsic distributions of GRBs due to their small sample size.Therefore,we update and enlarge the LGRB redshift sample detected by the Swift satellite in chapter 4,and use the above-mentioned maximum likelihood method to review the redshift and luminosity distributions of LGRBs.Given the incomplete sample,we have to correct the biases caused by the trigger problem and the redshift-measurement problem.Here we demonstrate that two simple sigmoid functions are enough to efficiently fit the trigger probability and redshift-measurement probability as functions of peak flux.After taking into account these two probabilities,we find that a strong redshift evolution in luminosity or density is still required in order to reproduce the observations well.However,different from previous works,the luminosity distribution is represented by a triple power-law function well here.Some studies which considered bright bursts only to enhance the completeness of sample can merely detect a broken power-law-like luminosity distribution.As the observed redshift distribution is affected by the luminosity threshold,the redshift and luminosity distributions were fitted simultaneously usually.If our goal is the empirical redshift distribution of GRBs,it is sufficient that only consider their luminosity distribution at the same time.However,if the rate of LGRBs is described by SFR which is an average results of over all galaxies,we should also take into account the bias of stellar mass distributions between LGRB host galaxies and whole galaxies to correctly study the evolution behaviours between LGRB rate and SFR.Therefore,in chapter 5,we consider a Schechter stellar mass function(SMF)for LGRB host galaxies to correct the bias between LGRB hosts and whole galaxies.We find that the simple star formation model still underestimates the number of LGRBs at high redshifts under the correction of this Schechter SMF,indicating that the LGRB rate and the SFR are indeed different.In addition,we consider the density evolution model,the energy(luminosity)evolution model and the mass evolution model to fit the observations of LGRBs.The results show that the density and mass evolution models can successfully describe the observed distributions of LGRBs,meanwhile larger sample of LGRB hosts is needed to distinguish these two models.Nonetheless,the expected distributions of energy evolution model are weekly different from the observed and its statistical probability is only～0.05.In previous works,the density evolution model and the energy evolution model were coupled together graphically and statistically,while show greater difference here.This hints the importance of correcting the bias between LGRB hosts and whole galaxies.