The Next Generation Virgo Cluster Survey has recently determined the luminosity function of galaxies in the core of the Virgo cluster down to unprecedented magnitude and surface brightness limits. Comparing simulations of cluster formation to the der
ived central stellar mass function, we attempt to estimate the stellar-to-halo-mass ratio (SHMR) for dwarf galaxies, as it would have been before they fell into the cluster. This approach ignores several details and complications, e.g., the contribution of ongoing star formation to the present-day stellar mass of cluster members, and the effects of adiabatic contraction and/or violent feedback on the subhalo and cluster potentials. The final results are startlingly simple, however; we find that the trends in the SHMR determined previously for bright galaxies appear to extend down in a scale-invariant way to the faintest objects detected in the survey. These results extend measurements of the formation efficiency of field galaxies by two decades in halo mass, or five decades in stellar mass, down to some of the least massive dwarf galaxies known, with stellar masses of $sim 10^5 M_odot$.
We use semi-analytic models of structure formation to interpret gravitational lensing measurements of substructure in galaxy cluster cores (R<=250kpc/h) at z=0.2. The dynamic range of the lensing-based substructure fraction measurements is well match
ed to the theoretical predictions, both spanning f_sub~0.05-0.65. The structure formation model predicts that f_sub is correlated with cluster assembly history. We use simple fitting formulae to parameterize the predicted correlations: Delta_90 = tau_90 + alpha_90 * log(f_sub) and Delta_50 = tau_50 + alpha_50 * log(f_sub), where Delta_90 and Delta_50 are the predicted lookback times from z=0.2 to when each theoretical cluster had acquired 90% and 50% respectively of the mass it had at z=0.2. The best-fit parameter values are: alpha_90 = (-1.34+/-0.79)Gyr, tau_90 = (0.31+/-0.56)Gyr and alpha_50 = (-2.77+/-1.66)Gyr, tau_50 = (0.99+/-1.18)Gyr. Therefore (i) observed clusters with f_sub<~0.1 (e.g. A383, A1835) are interpreted, on average, to have formed at z>~0.8 and to have suffered <=10% mass growth since z~0.4, (ii) observed clusters with f_sub>~0.4 (e.g. A68, A773) are interpreted as, on average, forming since z~0.4 and suffering >10% mass growth in the ~500Myr preceding z=0.2, i.e. since z=0.25. In summary, observational measurements of f_sub can be combined with structure formation models to estimate the age and assembly history of observed clusters. The ability to ``age-date approximately clusters in this way has numerous applications to the large clusters samples that are becoming available.
