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

Abstract

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 ($\mathrm{M}_V \lesssim -13$) is regulated primarily 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 are unable to 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 internal "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.

Figures (11)

• Figure 1: A comparison of satellite properties predicted by galform and observational data for the Milky Way (MW). In both panels, the results for 100 examples extracted from the Latin hypercube parameter space exploration are shown as thin copper lines, while the fiducial model is indicated by the thick solid curve of the same colour. These 100 runs were selected randomly from the full sample of 2600 runs; we find that these largely encompass the full range of predictions made by the model. Left panel: the satellite luminosity function, which shows the number of satellites as a function of absolute V-band magnitude, $\mathrm{M}_V$. The model predictions are compared against the combined satellite count for the Milky Way and M31, as described in Section \ref{['sec:emulator']}; these are shown as black triangles with error bars -- hereafter, we label this combined data set "MW satellites" for simplicity. The fiducial galform model provides a reasonable qualitative agreement with the observed satellite counts, particularly for brighter satellites ($\mathrm{M}_V < -10$). Right panel: the satellite mass-metallicity relation, with stellar metallicity ($Z_\star$) shown as a function of $\mathrm{M}_V$. galform predictions are compared with data compiled by Simon2019 for Milky Way satellites and the best-fit relation from Kirby2013 (dashed line, with the associated shaded region showing a 0.25 dex uncertainty around the median relation). The fiducial model predicts systematically lower metallicities at a fixed magnitude than what is observed in the data. In both panels, we find that galform models span a range of predictions, including many examples that are likely implausible and in strong tension with the data. The thin lines are matched across the left- and right-hand panels, and the shaded regions mark the 68% scatter around the median in the fiducial run.
• Figure 2: The validation loss as a function of training epoch for five distinct emulator models. In particular, we show the Mean Absolute Error (MAE), a combined metric that averages the loss from both the luminosity function and the mass-metallicity relation in some weighted combination. Each coloured line corresponds to an identical model architecture trained on a different fold of the training data in a 5-fold cross-validation scheme. We find that the validation error for all models decreases rapidly during the initial 50 epochs of training, after which performance plateaus as the models converge. All five models achieve a similar final MAE, reaching a value between 0.02 and 0.03 after $\sim$120 epochs.
• Figure 3: First-order Sobol' sensitivity indices, $S_1$, for the $z=0$ satellite luminosity function. Each pixel represents the sensitivity of the galform predictions in a bin of absolute V-band magnitude, $M_V$, to variations in each of the 11 input parameters of the model (see Section \ref{['sec:galform']} for their descriptions). The value of $S_1$, indicated by the adjoining colour scale, quantifies the fraction of the output variance in the luminosity function that is attributable directly to a single parameter defining the model. Higher values of $S_1$ correspond to parameters that are highly influential in the predicted abundance of satellite galaxies, while values close to zero represent parameters that have little to no influence. We find that the set of influential parameters changes as a function of satellite luminosity. The abundance of bright satellites ($M_V \lesssim -13$) is predominantly sensitive to parameters governing supernova feedback, specifically the parameters $\gamma_\mathrm{SN}$ and $V_{\mathrm{SN}}^{\mathrm{disk}}$, which define the mass loading factor. For fainter satellites, the sensitivity to the redshift and filtering scale for reionisation ($z_{\mathrm{reion}}$ and $V_\mathrm{crit}$, respectively) increases significantly. Parameters relating to mergers, quiescent star formation in disks and starbursts have no discernible impact on the final outcome.
• Figure 4: As Figure \ref{['fig:S1_SLF']}, but now showing the case of the satellite mass (luminosity)-metallicity relation.
• Figure 5: The first-order ($S_1$, shown as points with error bars) and total ($S_\mathrm{T}$, shown using solid lines) Sobol' sensitivity indices for various galform input parameters. For clarity, we show results for only the five most influential parameters in the model (based on the tests presented in Figures \ref{['fig:S1_SLF']} and \ref{['fig:S1_SMZ']}). Left panel: the sensitivity of the satellite luminosity function to these parameters. As noted previously, the parameters defining the supernova feedback scheme ($\gamma_\mathrm{SN}$ and $V_{\mathrm{SN}}^{\mathrm{disk}}$) are the most influential, although we observe considerable higher-order interaction with other parameters, as seen by the difference between the $S_1$ and $S_\mathrm{T}$ values of a given colour. Right panel: as in the left-hand panel, but now for the mass-metallicity relation for Milky Way satellites. In contrast to the luminosity function, the predicted metallicities are almost entirely governed by the physics of supernova feedback, which shows comparatively little higher-order interactions with other physical processes in shaping these predictions.