Best fits to the observational constraints from models with intrinsic E_p − L correlations and a luminosity function that does not evolve with redshift (k_evol = 0). The red curve corresponds to the case where the LGRB rate follows the shape of CSFRD and the blue curve corresponds to the case where the LGRB efficiency is free to evolve with redshift. The black curves and grey histograms are the observational constraints described in Fig. 2. This figure is discussed in Sect. 4.