An observational proxy of halo assembly time and its correlation with
  galaxy properties
  We show that the ratio between the stellar mass of central galaxy and the
mass of its host halo, $f_c \equiv M_{*,c}/M_{\rm h}$, can be used as an
observable proxy of halo assembly time, in that galaxy groups with higher $f_c$
assembled their masses earlier. Using SDSS groups of Yang et al., we study how
$f_c$ correlates with galaxy properties such as color, star formation rate,
metallicity, bulge to disk ratio, and size. Central galaxies of a given stellar
mass in groups with $f_c>0.02$ tend to be redder in color, more quenched in
star formation, smaller in size, and more bulge dominated, as $f_c$ increases.
The trends in color and star formation appear to reverse at $f_c<0.02$,
reflecting a down-sizing effect that galaxies in massive halos formed their
stars earlier although the host halos themselves assembled later (lower $f_c$).
No such reversal is seen in the size of elliptical galaxies, suggesting that
their assembly follows halo growth more closely than their star formation.
Satellite galaxies of a given stellar mass in groups of a given halo mass tend
to be redder in color, more quenched in star formation and smaller in size as
$f_c$ increases. For a given stellar mass, satellites also tend to be smaller
than centrals. The trends are stronger for lower mass groups. For groups more
massive than $\sim 10^{13}{\rm M}_\odot$, a weak reversed trend is seen in
color and star formation. The observed trends in star formation are
qualitatively reproduced by an empirical model based on halo age abundance
matching, but not by a semi-analytical model tested here.
