The upper bound on the lowest mass halo

We explore the connection between galaxies and dark matter halos in the Milky Way (MW) and quantify the implications on properties of the dark matter particle and the phenomenology of low-mass galaxy formation. This is done through a probabilistic comparison of the luminosity function of MW dwarf satellite galaxies to models based on two suites of zoom-in simulations. One suite is dark-matter-only while the other includes a disk component, therefore we can quantify the effect of the MW's baryonic disk on our results. We apply numerous Stellar-Mass-Halo-Mass (SMHM) relations allowing for multiple complexities: scatter, a characteristic break scale, and subhalos which host no galaxy. In contrast to previous works we push the model/data comparison to the faintest dwarfs by modeling observational incompleteness, allowing us to draw three new conclusions. Firstly, we constrain the SMHM relation for $10^2<M_*/M_\odot<10^8$ galaxies, allowing us to bound the peak halo mass of the faintest MW satellite to $M_\mathrm{vir}>2.4\times10^8M_\odot$ ($1\sigma$). Secondly, by translating to a Warm Dark Matter (WDM) cosmology, we bound the thermal relic mass $m_\mathrm{WDM}>2.9$ keV at 95\% confidence, on a par with recent constraints from the Lyman-$\alpha$ forest. Lastly, we find that the observed number of ultra-faint MW dwarfs is in tension with the theoretical prediction that reionisation prevents galaxy formation in almost all $10^8M_\odot$ halos. This can be tested with the next generation of deep imaging surveys. To this end, we predict the likely number of detectable satellite galaxies in the Subaru/HSC survey and the LSST. Confronting these predictions with future observations will be amongst our strongest tests of WDM and the effect reionisation on low-mass systems.