3D elemental abundances of stars at formation across the histories of
  Milky Way-mass galaxies in the FIRE simulations
  We characterize the 3-D 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 $\approx 7$ 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 $\gtrsim
12$ Gyr ago. The strength of the present-day abundance gradient does not
correlate with when the disk `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 degree scatter decreased
over time, from $\lesssim 0.17$ dex at $t_{\rm lb} = 11.6$ Gyr to $\sim 0.04$
dex at present day. Consequently, stars at $t_{\rm lb} \gtrsim 8$ Gyr formed in
a disk with primarily azimuthal scatter in abundances. All stars formed in a
vertically homogeneous disk, $\Delta$[Fe/H] $\leq 0.02$ dex within $1$ kpc of
the galactic midplane, with the exception of the young stars in the inner
$\approx 4$ kpc at $z \sim 0$. These results generally agree with our previous
analysis of gas-phase elemental abundances, which reinforces the importance of
cosmological disk evolution and azimuthal scatter in the context of stellar
chemical tagging. We provide analytic fits to our results for use in
chemical-tagging analyses.
