Observations of z∼6 quasars powered by super-massive black holes (SMBHs, MBH∼108−10M⊙) challenge our current understanding of early black hole formation and evolution. The advent of the James Webb Space Telescope (JWST) has enabled the study of massive black holes (MBHs, MBH∼106−7 M⊙) up to z∼11, thus bridging the properties of z∼6 quasars to their ancestors. JWST spectroscopic observations of GN-z11, a well-known z=10.6 star forming galaxy, have been interpreted with the presence of a super-Eddington (Eddington ratio ≡λEdd∼5.5) accreting MBH. To test this hypothesis we use a zoom-in cosmological simulation of galaxy formation and BH co-evolution. We first test the simulation results against the observed probability distribution function (PDF) of λEdd found in z∼6 quasars. Then, we select in the simulation those BHs that satisfy the following criteria: (a) 10<11, (b) MBH>106 M⊙. Finally we apply the Extreme Value Statistics to the PDF of λEdd resulting from the simulation and find that the probability of observing a z∼10−11 MBH, accreting with λEdd∼5.5, in the volume surveyed by JWST, is very low (<0.5%). We compare our predictions with those in the literature and further discuss the main limitations of our work. Our simulation cannot explain the JWST observations of GN-z11. This might be due to (i) missing physics in simulations, or (ii) uncertainties in the data analysis.
