A galactic tug-of-war: how (not) to simultaneously fit the Milky Way satellite luminosity function and the mass-metallicity relation

The satellite population of the Milky Way is shaped by a range of astrophysical processes including mergers, star formation, feedback, and cosmic reionisation. Determining which processes most strongly influence its properties is challenging and a key test of galaxy formation models. We train a neural network on the galform semi-analytic model and apply a variance-based sensitivity analysis to characterise the influence of 11 astrophysical parameters on two key observables: the satellite luminosity function, and the mass-metallicity relation. We find that: (1) the abundance of bright satellites (MV ≲ −13) is regulated by supernova feedback; (2) the faint end of the luminosity function is shaped by the interplay between feedback and reionisation; and (3) the mass-metallicity relation is governed almost exclusively by feedback at all masses. We do not find a combination of parameters in the fiducial model that fits the observed data for both statistics simultaneously. To understand why, we employ SHapley Additive exPlanations to capture the directionality of each parameter variation. This enables us to pinpoint the origin of tensions in the model, showing that parameter adjustments that regulate the abundance of faint satellites drive stellar metallicities to be an order of magnitude too low and vice versa. This “tug-of-war” leads us to consider extensions to the baseline model, such as metal loading in winds, or allowing the feedback strength to evolve with redshift. Our study highlights the value of interrogating complex physical models through a sensitivity analysis framework by revealing high-order parameter interactions and non-linear responses that traditional one-at-a-time variations would miss.