Constraining the intrinsic population of Long Gamma-Ray Bursts: implications for spectral correlations, cosmic evolution and their use as tracers of star formation

Long Gamma-Ray Bursts (LGRBs) have been shown to be powerful probes of the Universe, in particular to study the star formation rate up to very high redshift ($z \sim 9$). Since LGRBs are produced by only a small fraction of massive stars, it is paramount to have a good understanding of their underlying intrinsic population in order to use them as cosmological probes without introducing any unwanted bias. The goal of this work is to constrain and characterise this intrinsic population. We developed a Monte Carlo model where each burst is described by its redshift and its properties at the peak of the lightcurve. We derived the best fit parameters by comparing our synthetic populations to carefully selected observational constraints based on the CGRO/BATSE, Fermi/GBM and Swift/BAT samples with appropriate flux thresholds. We explored different scenarios in terms of cosmic evolution of the luminosity function and/or of the redshift distribution as well as including or not the presence of intrinsic spectral-energetics (Ep-L) correlations. We find that the existence of an intrinsic Ep-L correlation is preferred but with a shallower slope than observed($\alpha_A \sim 0.3$) and a larger scatter ($\sim 0.4$ dex). We find a strong degeneracy between the cosmic evolution of the luminosity and of the LGRB rate, and show that a sample both larger and deeper than SHOALS by a factor of three is needed to lift this degeneracy. The observed We conclude that Ep-L correlation cannot be explained only by selection effects although these do play a role in shaping the observed relation. The degeneracy between cosmic evolution of the luminosity function and of the redshift distribution of LGRBs should be included in the uncertainties of star formation rate estimates; these amount to a factor of 10 at $z=6$ and up to a factor of 50 at $z=9$.