On the scatter in the relation between stellar mass and halo mass: random or halo formation time dependent?

The empirical traditional halo occupation distribution (HOD) model of Wang et al. fits, by construction, both the stellar mass function and correlation function of galaxies in the local Universe. In contrast, the semi-analytical models of De Lucia & Blazoit (hereafter DLB07) and Guo et al. (hereafter Guo11), built on the same dark matter halo merger trees than the empirical model, still have difficulties in reproducing these observational data simultaneously. We compare the relations between the stellar mass of galaxies and their host halo mass in the three models, and find that they are different. When the relations are rescaled to have the same median values and the same scatter as in Wang et al., the rescaled DLB07 model can fit both the measured galaxy stellar mass function and the correlation function measured in different galaxy stellar mass bins. In contrast, the rescaled Guo11 model still overpredicts the clustering of low-mass galaxies. This indicates that the detail of how galaxies populate the scatter in the stellar mass-halo mass relation does play an important role in determining the correlation functions of galaxies. While the stellar mass of galaxies in the Wang et al. model depends only on halo mass and is randomly distributed within the scatter, galaxy stellar mass depends also on the halo formation time in semi-analytical models. At fixed value of infall mass, galaxies that lie above the median stellar mass-halo mass relation reside in haloes that formed earlier, while galaxies that lie below the median relation reside in haloes that formed later. This effect is much stronger in Guo11 than in DLB07, which explains the overclustering of low mass galaxies in Guo11. Assembly bias in Guo11 model might be overly strong. Nevertheless, in case that a significant assembly bias indeed exists in the real Universe, one needs to use caution when applying current HOD and abundance matching models that employ the assumption of random scatter in the relation between stellar and halo mass.