3D elemental abundances of stars at formation across the histories of Milky Way-mass galaxies in the FIRE simulations

We characterize the 3D spatial variations of [Fe/H], [Mg/H], and [Mg/Fe] in stars at the time of their formation, across 11 simulated Milky Way (MW)- and M31-mass galaxies in the FIRE-2 simulations, to inform initial conditions for chemical tagging. The overall scatter in [Fe/H] within a galaxy decreased with time until $\approx 7 \, \rm {Gyr}$ ago, after which it increased to today: this arises from a competition between a reduction of azimuthal scatter and a steepening of the radial gradient in abundance over time. The radial gradient is generally negative, and it steepened over time from an initially flat gradient $\gtrsim 12 \, \rm {Gyr}$ ago. The strength of the present-day abundance gradient does not correlate with when the disc 'settled'; instead, it best correlates with the radial velocity dispersion within the galaxy. The strength of azimuthal variation is nearly independent of radius, and the 360 deg scatter decreased over time, from $\lesssim 0.17 \, \rm {dex}$ at $t_{\rm lb} = 11.6 \, \rm {Gyr}$ to $\sim 0.04 \, \rm {dex}$ at present-day. Consequently, stars at $t_{\rm lb} \gtrsim 8 \, \rm {Gyr}$ formed in a disc with primarily azimuthal scatter in abundances. All stars formed in a vertically homogeneous disc, Δ[Fe/H]$\le 0.02 \, \rm {dex}$ within $1 \, \rm {kpc}$ of the galactic mid-plane, with the exception of the young stars in the inner $\approx 4 \, \rm {kpc}$ at z ~ 0. These results generally agree with our previous analysis of gas-phase elemental abundances, which reinforces the importance of cosmological disc evolution and azimuthal scatter in the context of stellar chemical tagging. We provide analytic fits to our results for use in chemical-tagging analyses.