| Abstract
| - Aims. We attempt to compile a complete, mass-selected sample of galaxies with low specific star-formation rates, and compare their properties with theoretical model predictions. Methods. We use the $f(24~{\rm \mu m})/f(K)$ flux ratio and the SED fitting to the $0.35{-}8.0~\mu$m spectral distribution, to select quiescent galaxies from $z\simeq 0.4$ to $z\simeq 4$ in the GOODS-MUSIC sample. Our observational selection can be translated into thresholds in specific star-formation rate $\dot{M}/M_*$, which can be compared with theoretical predictions. Results. In the framework of the well-known global decline in quiescent galaxy fraction with redshift, we find that a non-negligible fraction ${\simeq} 15{-}20\%$ of massive galaxies with low specific star-formation rate exists up to $z\simeq 4$, including a tail of “red and dead” galaxies with $\dot{M}/M_*<10^{-11}$ yr -1. Theoretical models vary to a large extent in their predictions for the fraction of galaxies with low specific star-formation rates, but are unable to provide a global match to our data.
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