An Empirical Model for the Star Formation History in Dark Matter Halos
  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
($>10^{12}M_{\odot}h^{-1}$) 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 $z_c\approx2$ above which the star formation
rate in low mass halos with masses $< 10^{11}M_{\odot}h^{-1}$ 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 $M_* < 10^8 M_{\odot}h^{-2}$ and steep slopes of high
redshift stellar mass and star formation rate functions.
