EDGE: the mass-metallicity relation as a critical test of galaxy
  formation physics
  We introduce the "Engineering Dwarfs at Galaxy formation's Edge" (EDGE)
project to study the cosmological formation and evolution of the smallest
galaxies in the Universe. In this first paper, we explore the effects of
resolution and sub-grid physics on a single low mass halo ($M_{\rm
halo}=10^{9}~M_\odot$), simulated to redshift $z=0$ at a mass and spatial
resolution of $\sim 20~M_\odot$ and $\sim 3$ pc. We consider different star
formation prescriptions, supernova feedback strengths and on-the-fly radiative
transfer (RT). We show that RT changes the mode of galactic self-regulation at
this halo mass, suppressing star formation by causing the interstellar and
circumgalactic gas to remain predominantly warm ($\sim 10^4$ K) even before
cosmic reionisation. By contrast, without RT, star formation regulation occurs
only through starbursts and their associated vigorous galactic outflows. In
spite of this difference, the entire simulation suite (with the exception of
models without any feedback) matches observed dwarf galaxy sizes, velocity
dispersions, $V$-band magnitudes and dynamical mass-to-light-ratios. This is
because such structural scaling relations are predominantly set by the host
dark matter halo, with the remaining model-to-model variation being smaller
than the observational scatter. We find that only the stellar mass-metallicity
relation differentiates the galaxy formation models. Explosive feedback ejects
more metals from the dwarf, leading to a lower metallicity at a fixed stellar
mass. We conclude that the stellar mass-metallicity relation of the very
smallest galaxies provides a unique constraint on galaxy formation physics.
