Organized chaos: scatter in the relation between stellar mass and halo mass in small galaxies

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 versus halo mass relation at low masses, M⋆ ≃ 10^5–10^8 M⊙. Assuming models with lognormal scatter about a median relation of the form M⋆ ∝ M^α_(halo), the preferred log-slope steepens from α ≃ 1.8 in the limit of zero scatter to α ≃ 2.6 in the case of 2 dex of scatter in M⋆ at fixed halo mass. We provide fitting functions for the best-fitting 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 (≳ 1 dex) and if the median relation is steep enough (α ≳ 2), then the ‘too-big-to-fail’ problem seen in the Local Group can be self-consistently eliminated in about ∼5–10 per cent of realizations. This scenario requires that the most massive subhaloes 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⋆ at fixed M_(halo) is large among different authors (∼2 dex), individual codes produce relations with much less scatter and usually give relations that would overproduce local galaxy counts.