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Cosmological simulations are reaching the resolution necessary to studyultra-faint dwarf galaxies. Observations indicate that in small populations,the stellar initial mass function (IMF) is not fully populated; rather, starsare sampled in a way that can be approximated as coming from an underlyingprobability density function. To ensure the accuracy of cosmologicalsimulations in the ultra-faint regime, we present an improved treatment of theIMF. We implement a self-consistent, stochastically populated IMF incosmological hydrodynamic simulations. We test our method using high-resolutionsimulations of a Milky Way halo, run to $z=6$, yielding a sample of nearly 100galaxies. We also use an isolated dwarf galaxy to investigate the resultingsystematic differences in galaxy properties. We find that a stochastic IMF insimulations makes feedback burstier, strengthening feedback, and quenching starformation earlier in small dwarf galaxies. For galaxies in halos with mass$\lesssim10^{8.5}$ M$_\odot$, a stochastic IMF typically leads to lower stellarmass compared to a continuous IMF, sometimes by more than an order ofmagnitude. We show that existing methods of ensuring discrete supernovaeincorrectly determine the mass of the star particle and its associatedfeedback. This leads to overcooling of surrounding gas, with at least${\sim}$10 per cent higher star formation and ${\sim}$30 per cent higher coldgas content. Going forward, to accurately model dwarf galaxies and compare toobservations, it will be necessary to incorporate a stochastically populatedIMF that samples the full spectrum of stellar masses.

Roll of the Dice: A Stochastically Sampled IMF Alters the Stellar Content of Simulated Dwarf Galaxies