Empirical derivation of the metallicity evolution with time and radius using TNG50 Milky Way/Andromeda analogues

Recent works have used a linear birth metallicity gradient to estimate the evolution of the [Fe/H] profile in the Galactic disk over time, and infer stellar birth radii (R$_\text{birth}$) from [Fe/H] and age measurements. These estimates rely on the evolution of [Fe/H] at the Galactic center ([Fe/H](0, $\tau$)) and the birth metallicity gradient ($\nabla$[Fe/H]($\tau)$) over time -- quantities that are unknown and inferred under key assumptions. In this work, we use the sample of Milky Way/Andromeda analogues from the TNG50 simulation to investigate the ability to recover [Fe/H](R, $\tau$) and R$_\text{birth}$ in a variety of galaxies. Using stellar disk particles, we test the assumptions required in estimating R$_\text{birth}$, [Fe/H](0, $\tau$), and $\nabla$[Fe/H]($\tau)$ using recently proposed methods to understand when they are valid. We show that $\nabla$[Fe/H]($\tau)$ can be recovered in most galaxies to within 26% from the range in [Fe/H] across age, with better accuracy for more massive and stronger barred galaxies. We also find that the true central metallicity is unrepresentative of the genuine disk [Fe/H] profile; thus we propose to use a projected central metallicity instead. About half of the galaxies in our sample do not have a continuously enriching projected central metallicity, with a dilution in [Fe/H] correlating with mergers. Most importantly, galaxy-specific [Fe/H](R, $\tau$) can be constrained and confirmed by requiring the R$_\text{birth}$ distributions of mono-age, solar neighborhood populations to follow inside-out formation. We conclude that examining trends with R$_\text{birth}$ is valid for the Milky Way disk and similarly structured galaxies, where we expect R$_\text{birth}$ can be recovered to within 20% assuming today's measurement uncertainties in TNG50.