  We develop a simple analytical model that tracks galactic metallicities
governed by star formation and feedback to gain insight from the observed
galaxy stellar mass-metallicity relations over a large range of stellar masses
and redshifts. The model reveals the following implications of star formation
and feedback processes in galaxy formation. First, the observed metallicity
relations provide a stringent upper limit for the averaged outflow mass-loading
factors of local galaxies, which is ~20 for M_*~10^9Msun galaxies and
monotonically decreases to ~1 for M_*~10^{11}Msun galaxies. Second, the
inferred upper-limit for the outflow mass-loading factor sensitively depends on
whether the outflow is metal-enriched with respect to the ISM metallicity. If
half of the metals ejected from SNe leave the galaxy in metal-enriched winds,
the outflow mass-loading factor for galaxies at any mass can barely be higher
than ~10, which puts strong constraints on galaxy formation models. Third, the
relatively lower stellar-phase to gas-phase metallicity ratio for lower-mass
galaxies indicate that low-mass galaxies are still rapidly enriching their
metallicities in recent times, while high-mass galaxies are more settled, which
seems to show a downsizing effect in the metallicity evolution of galaxies. The
analysis presented in the paper demonstrates the importance of accurate
measurements of galaxy metallicities and the cold gas fraction of galaxies at
different redshifts for constraining star formation and feedback processes, and
demonstrates the power of these relations in constraining the physics of galaxy
formation.
