Predicting Quiescence: The Dependence of Specific Star Formation Rate on Galaxy Size and Central Density at 0.5<z<2.5

In this paper, we investigate the relationship between star formation and structure, using a mass-complete sample of 27,893 galaxies at $0.5<z<2.5$ selected from 3D-HST. We confirm that star-forming galaxies are larger than quiescent galaxies at fixed stellar mass (M$_{\star}$). However, in contrast with some simulations, there is only a weak relation between star formation rate (SFR) and size within the star-forming population: when dividing into quartiles based on residual offsets in SFR, we find that the sizes of star-forming galaxies in the lowest quartile are 0.27$\pm$0.06 dex smaller than the highest quartile. We show that 50% of star formation in galaxies at fixed M$_{\star}$ takes place within a narrow range of sizes (0.26 dex). Taken together, these results suggest that there is an abrupt cessation of star formation after galaxies attain particular structural properties. Confirming earlier results, we find that central stellar density within a 1 kpc fixed physical radius is the key parameter connecting galaxy morphology and star formation histories: galaxies with high central densities are red and have increasingly lower SFR/M$_{\star}$, whereas galaxies with low central densities are blue and have a roughly constant (higher) SFR/M$_{\star}$ at a given redshift. We find remarkably little scatter in the average trends and a strong evolution of $>$0.5 dex in the central density threshold correlated with quiescence from $z\sim0.7-2.0$. Neither a compact size nor high-$n$ are sufficient to assess the likelihood of quiescence for the average galaxy; rather, the combination of these two parameters together with M$_{\star}$ results in a unique quenching threshold in central density/velocity.