| Abstract
| - Context. To answer questions on the start and duration of the epoch of reionisation, periods of galaxy mergers and properties of other cosmological encounters, the cosmic star formation history, \hbox{$\dot{\rho}_{*}$}ρ̇∗ or CSFH, is of fundamental importance. Using the association of long-duration gamma-ray bursts (LGRBs) with the death of massive stars and their ultra-luminous nature (>10 52ergs -1), the CSFH can be probed to higher redshifts than current conventional methods. Unfortunately, no consensus has been reached on the manner in which the LGRB rate, \hbox{$\dot{\rho}_{\rm grb}$}ρ̇grb or LGRBR, traces the CSFH, leaving many of the questions mentioned mostly unexplored by this method. Aims. Observations by the gamma-ray burst near-infrared detector (GROND) over the past 4 years have, for the first time, acquired highly complete LGRB samples. Driven by these completeness levels and new evidence of LGRBs also occurring in more massive and metal rich galaxies than previously thought, the possible biases of the \hbox{$\dot{\rho}_{\rm grb}$}ρ̇grb- \hbox{$\dot{\rho}_{*}$}ρ̇∗ connection are investigated over a large range of galaxy properties. Methods. The CSFH is modelled using empirical fits to the galaxy mass function and galaxy star formation rates. Biasing the CSFH by means of metallicity cuts, mass range boundaries, and other unknown redshift dependencies of the form \hbox{$\dot{\rho}_{\rm grb}\propto\dot{\rho}_{*}(1+z)^{\delta}$}ρ̇grb ∝ ρ̇∗ (1 + z) δ, a \hbox{$\dot{\rho}_{\rm grb}$}ρ̇grb is generated and compared to the highly complete GROND LGRB sample. Results. It is found that there is no strong preference for a metallicity cut or fixed galaxy mass boundaries and that there are no unknown redshift effects ( δ = 0), in contrast to previous work which suggest values of Z/ Z⊙ ~ 0.1−0.3. From the best-fit models obtained, we predict that ~1.2% of the LGRB burst sample exists above z = 6. Conclusions. The linear relationship between \hbox{$\dot{\rho}_{\rm grb}$}ρ̇grb and \hbox{$\dot{\rho}_{*}$}ρ̇∗ suggested by our results implies that redshift biases present in previous LGRB samples significantly affect the inferred dependencies of LGRBs on their host galaxy properties. Such biases can lead to, for example, an interpretation of metallicity limitations and evolving LGRB luminosity functions.
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