The shutdown of star formation in galaxies is generally termed `quenching. Although quenching may occur through a variety of processes, the exact mechanism(s) that is in fact responsible for quenching is still in question. This paper addresses quenching by searching for traces of possible quenching processes through their effects on galaxy structural parameters such as surface stellar mass density and Sersic index (n). We analyze the rest-frame U-B color correlations versus these structural parameters using a sample of galaxies in the redshift range 0.5< z<0.8 from the DEEP2/AEGIS survey. We find that Sersic index (n) has the smallest overlap region among all tested parameters and resembles a step-function with a threshold value of n=2.3. There exists, however, a significant population of outliers with blue colors yet high n values that seem to contradict this behavior. We hypothesize that their Sersic values may be distorted by bursts of star formation, AGNs, and/or poor fits, leading us to consider central surface stellar mass density as an alternative to Sersic index. Not only does it correct the outliers, it also forms a tight relationship with color, suggesting that the innermost structure of galaxies is most physically linked with quenching. Furthermore, at z~0.65, the majority of the blue cloud galaxies cannot simply fade onto the red sequence since their GIM2D bulge masses are only half as large on average as the bulge masses of similar red sequence galaxies, thus demonstrating that stellar mass must absolutely increase at the centers of galaxies as they quench. We discuss a two-stage model for quenching in which galaxy star formation rates are controlled by their dark halos while they are still in the blue cloud and a second quenching process sets in later, associated with the central stellar mass build-up.
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