We studied the star formation rate (SFR) in cosmological hydrodynamical simulations of galaxy (proto-)clusters in the redshift range $0<z<4$, comparing them to recent observational studies; we also investigated the effect of varying the parameters of the star formation model on galaxy properties such as SFR, star-formation efficiency, and gas fraction. We analyze a set of zoom-in cosmological hydrodynamical simulations centred on twelve clusters. The simulations are carried out with the GADGET-3 TreePM/SPH code which includes various subgrid models to treat unresolved baryonic physics, including AGN feedback. Simulations do not reproduce the high values of SFR observed within protoclusters cores, where the values of SFR are underpredicted by a factor $gtrsim 4$ both at $zsim2$ and $zsim 4$. The difference arises as simulations are unable to reproduce the observed starburst population and is worsened at $zsim 2$ because simulations underpredict the normalization of the main sequence of star forming galaxies (i.e., the correlation between stellar mass and SFR) by a factor of $sim 3$. As the low normalization of the main sequence seems to be driven by an underestimated gas fraction, it remains unclear whether numerical simulations miss starburst galaxies due to a too low predicted gas fractions or too low star formation efficiencies. Our results are stable against varying several parameters of the star formation subgrid model and do not depend on the details of the AGN feedback.
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