Probing cold gas with Mg II and Ly$\alpha$ radiative transfer

The Mg II resonance doublet at 2796 {\AA} and 2803 {\AA} is an increasingly important tool to study cold, $T \sim 10^{4}\,$K, gas -- an observational driven development requiring theoretical support. We develop a new Monte Carlo radiative transfer code to systematically study the joined Mg II and Ly$\alpha$ escape through homogeneous and `clumpy' multiphase gas with dust in arbitrary 3D geometries. Our main findings are: (i) The Mg II spectrum differs from Ly$\alpha$ due to the large difference in column densities, even though the atomic physics of the two lines are similar. (ii) the Mg II escape fraction is generally higher than that of Ly$\alpha$ because of lower dust optical depths and path lengths -- but large variations due to differences in dust models and the clumpiness of the cold medium exist. (iii) Clumpy media possess a `critical covering factor' above which Mg II radiative transfer matches a homogeneous medium. The critical covering factors for Mg II and Ly$\alpha$ differ, allowing constraints on the cold gas structure. (iv) The Mg II doublet ratio $R_{\rm MgII}$ varies for strong outflows/inflows ($\gtrsim 700 \mathrm{km\,s}^{-1}$), in particular, $R_{\rm MgII}<1$ being an unambiguous tracer for powerful galactic winds. (v) Scattering of stellar continuum photons can decrease $R_{\rm MgII}$ from two to one, allowing constraints on the scattering medium. Notably, we introduce a novel probe of the cold gas column density -- the halo doublet ratio -- which we show to be a powerful indicator of ionizing photon escape. We discuss our results in the context of interpreting and modeling observations as well as their implications for other resonant doublets.