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Abstract We predict the stellar mass–halo mass (SMHM) relationship for dwarf galaxies, using simulated galaxies with peak halo masses of M peak = 10 11 M ⊙ down into the ultra-faint dwarf range to M peak = 10 7 M ⊙ . Our simulated dwarfs have stellar masses of M star = 790 M ⊙ to 8.2 × 10 8 M ⊙ , with corresponding V -band magnitudes from −2 to −18.5. For M peak > 10 10 M ⊙ , the simulated SMHM relationship agrees with literature determinations, including exhibiting a small scatter of 0.3 dex. However, the scatter in the SMHM relation increases for lower-mass halos. We first present results for well-resolved halos that contain a simulated stellar population, but recognize that whether a halo hosts a galaxy is inherently mass resolution dependent. We thus adopt a probabilistic model to populate “dark” halos below our resolution limit to predict an “intrinsic” slope and scatter for the SMHM relation. We fit linearly growing log-normal scatter in stellar mass, which grows to more than 1 dex at M peak = 10 8 M ⊙ . At the faintest end of the SMHM relation probed by our simulations, a galaxy cannot be assigned a unique halo mass based solely on its luminosity. Instead, we provide a formula to stochastically populate low-mass halos following our results. Finally, we show that our growing log-normal scatter steepens the faint-end slope of the predicted stellar mass function. 

ABSTRACT Cosmological simulations are reaching the resolution necessary to study ultra-faint dwarf galaxies. Observations indicate that in small populations, the stellar initial mass function (IMF) is not fully populated; rather, stars are sampled in a way that can be approximated as coming from an underlying probability density function. To ensure the accuracy of cosmological simulations in the ultra-faint regime, we present an improved treatment of the IMF. We implement a self-consistent, stochastically populated IMF in cosmological hydrodynamic simulations. We test our method using high-resolution simulations of a Milky Way halo, run to z = 6, yielding a sample of nearly 100 galaxies. We also use an isolated dwarf galaxy to investigate the resulting systematic differences in galaxy properties. We find that a stochastic IMF in simulations makes feedback burstier, strengthening feedback, and quenching star formation earlier in small dwarf galaxies. For galaxies in haloes with mass ≲ 108.5 M⊙, a stochastic IMF typically leads to lower stellar mass compared to a continuous IMF, sometimes by more than an order of magnitude. We show that existing methods of ensuring discrete supernovae incorrectly determine the mass of the star particle and its associated feedback. This leads to overcooling of surrounding gas, with at least ∼10 per cent higher star formation and ∼30 per cent higher cold gas content. Going forwards, to accurately model dwarf galaxies and compare to observations, it will be necessary to incorporate a stochastically populated IMF that samples the full spectrum of stellar masses.