The metallicity and star formation activity of long gamma-ray burst
  hosts for z$<$3: insights from the Illustris simulation
  We study the properties of long gamma-ray bursts (LGRBs) using a large scale
hydrodynamical cosmological simulation, the Illustris simulation. We determine
the LGRB host populations under different thresholds for the LGRB progenitor
metallicities, according to the collapsar model. We compare the simulated
sample of LGRBs hosts with recent, largely unbiased, host samples: BAT6 and
SHOALS. We find that at $z<1$ simulated hosts follow the mass-metallicity
relation and the fundamental metallicity relation simultaneously, but with a
paucity of high-metallicity hosts, in accordance with observations. We also
find a clear increment in the mean stellar mass of LGRB hosts and their SFR
with redshift up to $z<3$ on account of the metallicity dependence of
progenitors. We explore the possible origin of LGRBs in metal rich galaxies,
and find that the intrinsic metallicity dispersion in galaxies could explain
their presence. LGRB hosts present a tighter correlation between galaxy
metallicity and internal metallicity dispersion compared to normal star forming
galaxies. We find that the Illustris simulations favours the existence of a
metallicity threshold for LGRB progenitors in the range 0.3 - 0.6 Z$_\odot$
