  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 $z\sim2$ and $z\sim 4$. The difference arises as
simulations are unable to reproduce the observed starburst population and is
worsened at $z\sim 2$ 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.
