The relation between star formation rate and stellar mass of galaxies at
  z $\sim$ 1-4
  The relation between the Star Formation Rate (SFR) and stellar mass (${\rm
M}_{\star}$) of galaxies represents a fundamental constraint on galaxy
formation and has been studied extensively both in observations and
cosmological simulations. However, the observed amplitude has not been
successfully reproduced in simulations, indicating either that the halo
accretion history and baryonic physics are poorly modeled or that observations
contain biases. We examine the evolution of the SFR$-{\rm M}_{\star}$ relation
of $z\sim1-4 $ galaxies and display the inconsistency between observed
relations that are obtained using different techniques. We employ cosmological
hydrodynamic simulations and compare these with a range of observed SFR$-{\rm
M}_{\star}$ relations. We find that numerical results are consistent with
observations that use Spectral Energy Distribution (SED) techniques to estimate
star formation rates and dust corrections. On the contrary, simulations are not
able to reproduce results that were obtained by combining only UV and IR
luminosities. These imply SFRs at a fixed stellar mass that are larger almost
by a factor of 5 than those of SED measurements for $z \sim1.5-4$. Furthermore,
we find remarkable agreement between the numerical results from various authors
who have employed different cosmological codes and run simulations with
different resolutions. This is interesting for two reasons. A) simulations can
produce realistic populations of galaxies within representative cosmological
volumes even at relatively modest resolutions. B) It is likely that current
numerical codes that rely on similar subgrid multiphase Inter-Stellar Medium
(ISM) models and are tuned to reproduce statistical properties of galaxies,
produce similar results for the SFR$-{\rm M}_{\star}$ relation by construction,
regardless of resolution, box size and, to some extent, the adopted feedback
prescriptions.
