The relationship between age, metallicity, and abundances for disk stars
  in a simulated Milky Way galaxy
  Observations of the Milky Way's low-$\alpha$ disk show that at fixed
metallicity, [Fe/H], several element abundance, [X/Fe], correlate with age,
with unique slopes and small scatters around the age-[X/Fe] relations. In this
study, we turn to simulations to explore the age-[X/Fe] relations for the
elements C, N, O, Mg, Si, S, and Ca that are traced in a FIRE-2 cosmological
zoom-in simulation of a Milky Way-like galaxy, m12i, and understand what
physical conditions give rise to the observed age-[X/Fe] trends. We first
explore the distributions of mono-age populations in their birth and current
locations, [Fe/H], and [X/Fe], and find evidence for inside-out radial growth
for stars with ages < 7 Gyr. We then examine the age-[X/Fe] relations across
m12i's disk and find that the direction of the trends agree with observations,
apart from C, O, and Ca, with remarkably small intrinsic scatters,
$\sigma_{int}$ (0.01-0.04 dex). This $\sigma_{int}$ measured in the simulations
is also metallicity-dependent, with $\sigma_{int}$ $\approx$ 0.025 dex at
[Fe/H]=-0.25 dex versus $\sigma_{int}$ $\approx$ 0.015 dex at [Fe/H]=0 dex, and
a similar metallicity dependence is seen in the GALAH survey for the elements
in common. Additionally, we find that $\sigma_{int}$ is higher in the inner
galaxy, where stars are older and formed in less chemically-homogeneous
environments. The age-[X/Fe] relations and the small scatter around them
indicate that simulations capture similar chemical enrichment variance as
observed in the Milky Way, arising from stars sharing similar element
abundances at a given birth place and time.
