The environment of Long Gamma-Ray Bursts revealed by their host galaxies 

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Scientific landscape for GRB population models

Population models have become a popular approach to study GRBs due to the increasing sample size and the affordable computing power, here I review a few notable models which influenced the development of our project. It should be noted that the publication date of these different works spans 17 years, during which GRB samples have improved in size and accuracy alongside our knowledge of the star-formation rate and more generally our understanding of the Universe.
The basic idea behind a population model is to simulate an intrinsic population of GRBs from certain distributions (luminosity, redshift, spectrum…) and then apply some detection criteria to create an observed sample; this falls in the category of a forward folding approach because one starts by simulating the data to then reproduce the observations. In some instances (e.g. Salvaterra et al., 2012; Pescalli et al., 2016), a different approach is used where the starting point of the analysis is the observed data itself and then some statistical methods are used to recover the intrinsic population from the observed one. In both approaches however, the results are expected to be the same. The main results regarding the luminosity function and redshift distributions are compiled in Fig. 2.2 and 2.3.

Porciani & Madau (2001), (P01)

This is one of the most notable early works on the redshift distribution of GRBs. They used a power law-like, non-evolving luminosity function with a cut-off at low luminosities and due to the association of LGRBs with massive stars, they used a propotionality between the LGRB rate and the Cosmic Star Formation Rate Density (CSFRD). They used a standard Band function (see Sect. 2.2.3) with fixed parameters = −1, = −2.25 and Ep= 511 keV (in the source frame) for the GRB spectrum. The main constraint they used is the logN-logP diagram (see Sect. 4.1.1)  from BATSE/CGRO, corrected for efficiency by Kommers et al. (2000). At the time of the study the number of GRBs with redshift was very small, so they used the association of LGRBs with  assive stars to predict rates. It should also be noted that the CSFRD at high redshift was very uncertain in 2001, so the authors used three different scenarios, an increasing, constant and decreasing CSFRD beyond z 2.

Daigne, Rossi & Mochkovitch (2006), (D06)

This is the earlier version of our own population model. The authors used a power law, nonevolving luminosity function, with three different scenarios for the redshift distribution with different behaviors at high redshift largely inspired from P01: increasing, constant and decreasing above z 2. Since the CSFRD was more precisely determined than in P01 by 2006, this translates to different scenarios for the LGRB efficiency, i.e. an strongly increasing, mildy increasing or constant LGRB efficiency for each respective scenario. They assume a Band function for the GRB spectra with a Log-Normal scenario and an « Amati-like » scenario (see Sect. 3.2.2) for the Ep distribution and empirically-based distributions for and . Despite not having a full-fledged redshift constraint, they use the pioneering work of Jakobsson et al. (2006) based on the early data from Swift as a cross-check for the redshift distribution of their model and conclude that the data favors a model with a strongly increasing LGRB efficiency with high redshift using constraints from CGRO and HETE2. In other words, these results favored the hypothesis that LGRBs do not trace star-formation. However, it should be noted that they do not consider the possibility of a redshift evolution of the luminosity function.

Wanderman & Piran (2010), (WP10)

This is a study based on a sample of Swift LGRBs from which the luminosity function and the redshift distribution are estimated using a method ressembling the C− method (Lynden-Bell, 1971). The authors invert the observed redshift-luminosity distribution assuming a Band GRB spectrum with = −1, = −2.25 and Ep= 511 keV. They use a broken power law, non-evolving luminosity function and redshift distribution and conclude that there may be hints that the LGRB rate does not follow the CSFRD but the results are inconclusive. It is worth noting that they used all Swift bursts with a redshift but in order to avoid biases they derive a redshift measurement probability as a function of the peak photon flux whose robustness remains to be proven.

Salvaterra et al. (2012), (S12)

This study focused on the complete BAT6 sample defined in Sect. 4.3.2. They perform an analysis of the luminosity function and redshift distribution using a sample of 58 bright LGRBs from Swift with a selection on the peak photon flux of N > 2.6 ph s−1 cm−2 and use a statistical method to recover the intrinsic population distributions from the observed ones. They use two different function forms for the luminosity function: a power law with a cut-off at low luminosity (the same as P01) and a broken power law (same as WP10). In both cases they explore the possibiliy of a redshift evolution of the luminosity function, parametrized as (1 + z)kevol . For the redshift distribution they assume the LGRB rate follows the CSFRD but also allow for some evolution parametrized as (1 + z)nevol . They assume a Band GRB spectrum with = −1, = −2.25 and Ep drawn from the « Amati-like » scenario, assuming fixed values for the correlation. They conclude that an evolution of the luminosity function is degenerate with an evolution of the redshift distribution (i.e. an LGRB rate not following the CSFRD), but one or the other is required to reproduce their observed sample. It should be noted however that this work is based on fairly bright LGRBs, which is only the tip of the iceberg of the logN-logP diagram (see Sect. 4.1.1) This work is not presented in Tab. 2.2 in favor of P16 since the latter is an updated version.

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Parametrization of the intrinsic long GRB population

In this section, I present the functional forms used to describe the various distributions of our population model. Most often these were taken as simple as possible, when no compelling physical argument could be made in favor of a specific model.

Peak Luminosity Function

One of the most well-studied distribution is the Luminosity Function (LF) of LGRBs. Stricly speaking, it is the distribution of isotropic-equivalent luminosities Liso, which is defined as the luminosity the source would have if it emitted in 4 of the sky. Of course to measure this quantity one needs to know the distance to the GRB, making the number of bursts with a measure Liso a small fraction of the total number of bursts ever detected. An important point to keep mind is that the rates are always given for GRB pointing towards us; to get the entire rate of GRBs one needs to multiply the observed rate by 4 h i , where h i is the average opening angle. The real L distribution depends on the distribution of opening angles for the GRB jet. In the rest of the manuscript, the subscript iso will be omitted for ease of reading.

Redshift distribution and LGRB efficiency

As mentioned in Sect. 4.3, despite being a crucial property, the majority of LGRBs do not have a measured redshift. Therefore their intrinsic redshift distribution is uncertain, with different parametric forms and statistical methods used (e.g. Daigne et al., 2006; Wanderman & Piran, 2010; Yu et al., 2015; Petrosian et al., 2015; Pescalli et al., 2016; Amaral-Rogers et al., 2016), leading to conflicting results. The question of the redshift distribution of LGRBs is of vital importance for understanding the LGRB efficiency so we tried various parametric forms in our population model, described below after a formal definition of the LGRB efficiency.

Table of contents :

I General introduction 
1 Introduction 
1.1 Gamma-ray bursts
1.1.1 Main observational facts
1.1.2 Theoretical framework
1.2 The progenitors of long GRBs
1.3 Use of GRBs in cosmology
1.4 Structure of the manuscript
II A population model for Long Gamma-Ray Bursts 
2 Introduction
2.1 Scientific motivation
2.2 GRBs in the domain
2.2.1 Past, current (and future) missions
2.2.2 Distance
2.2.3 GRB spectrum
2.2.4 Fluxes in the domain
2.2.5 Fluence
2.3 Scientific landscape for GRB population models
3 Constructing an LGRB population 
3.1 Forward modeling approach
3.2 Parametrization of the intrinsic long GRB population
3.2.1 Peak Luminosity Function
3.2.2 Peak Energy distribution
3.2.3 Spectral slopes distribution
3.2.4 Redshift distribution and LGRB efficiency
3.2.5 Parametrization of the LGRB comoving rate
3.3 Creating mock samples
3.3.1 From the source frame to the observer frame
3.3.3 Fermi/GBM and Swift/BAT
3.4 Summary
4 Observational constraints 
4.1 Intensity Constraint
4.1.1 logN-logP
4.1.2 Efficiency correction
4.1.3 Normalization of the LGRB population
4.2 Spectral Constraint
4.2.1 Observed Ep distribution
4.3 Redshift Constraint
4.3.1 Biases in redshift distributions
4.3.2 BAT6: a well-controlled, complete sample
4.4 Additional observables for crosschecking and complementary studies
4.4.1 The eBAT6 Ep-L plane
4.4.2 The SHOALS redshift distribution
4.5 Summary
5 Parameter space exploration 
5.1 Statistical tools
5.1.1 Bayesian inference
5.1.2 Goodness of fit estimator
5.2 Exploring the parameter space
5.2.1 Brute Force
5.2.2 Markov Chain Mont Carlo
6 Results and discussion 
6.1 Introduction
6.2 Methodological check
6.3 Scenario without intrinsic spectrum-luminosity correlation
6.3.1 Scenarios with a constant LGRB efficiency (n˙ GRB / SFR)
6.3.2 Scenarios with a redshift distribution free to vary
6.3.3 Summary for LN-Ep models
6.4 Scenario with intrinsic spectrum-luminosity correlation
6.4.1 Scenarios with a constant LGRB efficiency (n˙ GRB / SFR)
6.4.2 Scenarios with an LGRB efficiency free to vary
6.4.3 Summary for A-Ep models
6.5 Discussion
6.5.1 Distinguishing between well-fitting scenarios
6.5.2 The intrinsic redshift distribution and the production efficiency of LGRBs
6.5.3 What is the luminosity function?
6.5.4 What is the true LGRB rate?
6.5.5 Is there an intrinsic « Amati-like » correlation?
6.6 Conclusion
Appendix A Results from population model MCMC exploration 
Appendix B Fits to the observational constraints from our population model 
Appendix C Additional cross-checks 
Appendix D The SVOM/ECLAIRs sample 
III The environment of Long Gamma-Ray Bursts revealed by their host galaxies 
7 Introduction 
7.1 Why study the host galaxies of LGRBs?
7.2 The BAT6 LGRB host sample
7.3 The X-Shooter instrument
8 Are LGRBs biased tracers of star formation? Clues from the host galaxies of the Swift/BAT6 complete sample of bright LGRBs (Palmerio, Vergani et al. 2018) 
8.1 Abstract
8.2 Introduction
8.3 The BAT6 sample of LGRB host galaxies at z > 1
8.3.1 Selection
8.3.2 Stellar mass
8.3.3 Star Formation Rate and Metallicity
8.4 Comparison with the star-forming galaxy population
8.4.1 Comparison samples
8.4.2 Bayesian framework
8.4.3 Stellar Mass
8.4.4 Star Formation Rate
8.4.5 Metallicity
8.5 Discussion
8.6 Conclusions
Appendix E Spectral Energy Distribution fitting with BEAGLE 
Appendix F LGRB host galaxies: magnitudes and emission line fluxes 
IV General conclusion 
9 General conclusion 
9.1 Summary of our main results
9.1.1 Population model for LGRBs
9.1.2 Host galaxies of LGRBs
9.2 Consequences for LGRB production efficiency
9.3 Perspectives
9.3.1 Extending the host galaxy study
9.3.2 Predictions for SVOM/ECLAIRs
9.3.3 Parametrization of the redshift distribution of the intrinsic LGRB population
9.3.4 Is our model representative of the whole intrinsic GRB population?
9.3.5 Can we extend the model to other GRB properties?


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