On halo formation times and assembly bias

In this paper, we use the 'Millennium Simulation' to re-examine the mass assembly history of dark matter haloes and the age dependence of halo clustering. We use eight different definitions of halo formation times to characterize the different aspects of the assembly history of a dark matter halo. We find that these formation times have different dependence on halo mass. While some formation times characterize well the hierarchical nature of halo formation, in the sense that more massive haloes have later formation, the trend is reversed for other definitions of the formation time. In particular, the formation times that are likely to be related to the formation of galaxies in dark haloes show strong trends of 'downsizing', in that lower mass haloes form later. We also investigate how the correlation amplitude of dark matter haloes depends on the different formation times. We find that this dependence is quite strong for some definitions of formation time but weak or absent for other definitions. In particular, the correlation amplitude of haloes of a given mass is almost independent of their last major merger time. For the definitions that are expected to be more related to the formation of galaxies in dark haloes, a significant assembly bias is found only for haloes less massive than M-*. We discuss our results in connection to the hierarchical assembly of dark matter haloes, the 'archaeological downsizing' observed in the galaxy population, and the observed colour-dependence of the clustering strength of galaxy groups and clusters.