We use Local Group galaxy counts together with the ELVIS N-body simulations to explore the relationship between the scatter and slope in the stellar mass vs. halo mass relation at low masses , M _ { \star } \simeq 10 ^ { 5 } -10 ^ { 8 } M _ { \odot } .
Assuming models with log-normal scatter about a median relation of the form M _ { \star } \propto M _ { \mathrm { halo } } ^ { \alpha } , the preferred log-slope steepens from \alpha \simeq 1.8 in the limit of zero scatter to \alpha \simeq 2.6 in the case of 2 dex of scatter in M _ { \star } at fixed halo mass .
We provide fitting functions for the best-fit relations as a function of scatter , including cases where the relation becomes increasingly stochastic with decreasing mass .
We show that if the scatter at fixed halo mass is large enough ( \gtrsim 1 dex ) and if the median relation is steep enough ( \alpha \gtrsim 2 ) , then the “ too-big-to-fail ” problem seen in the Local Group can be self-consistently eliminated in about \sim 5 - 10 \% of realizations .
This scenario requires that the most massive subhalos host unobservable ultra-faint dwarfs fairly often ; we discuss potentially observable signatures of these systems .
Finally , we compare our derived constraints to recent high-resolution simulations of dwarf galaxy formation in the literature .
Though simulation-to-simulation scatter in M _ { \star } at fixed M _ { \mathrm { halo } } is large among separate authors ( \sim 2 dex ) , individual codes produce relations with much less scatter and usually give relations that would over-produce local galaxy counts .