An empirical model for the star formation history in dark matter haloes

We develop an empirical approach to infer the star formation rate in dark matter halos from the galaxy stellar mass function (SMF) at different redshifts and the local cluster galaxy luminosity function (CGLF), which has a steeper faint end relative to the SMF of local galaxies. As satellites are typically old galaxies which have been accreted earlier, this feature can cast important constraint on the formation of low-mass galaxies at high-redshift. The evolution of the SMFs suggests the star formation in high mass halos (>10^12M_h^-1) has to be boosted at high redshift beyond what is expected from a simple scaling of the dynamical time. The faint end of the CGLF implies a characteristic redshift zc2 above which the star formation rate in low mass halos with masses < 10^11M_h^-1 must be enhanced relative to that at lower z. This is not directly expected from the standard stellar feedback models. Also, this enhancement leads to some interesting predictions, for instance, a significant old stellar population in present-day dwarf galaxies with M_* < 10⁸ M_h^-2 and steep slopes of high redshift stellar mass and star formation rate functions.