Forecasting Chemical Abundance Precision for Extragalactic Stellar
  Archaeology
  Increasingly powerful and multiplexed spectroscopic facilities promise
detailed chemical abundance patterns for millions of resolved stars in galaxies
beyond the Milky Way (MW). Here, we employ the Cram\'er-Rao Lower Bound (CRLB)
to forecast the precision to which stellar abundances for metal-poor, low-mass
stars outside the MW can be measured for 41 current (e.g., Keck, MMT, VLT,
DESI) and planned (e.g., MSE, JWST, ELTs) spectrograph configurations. We show
that moderate resolution ($R\lesssim5000$) spectroscopy at blue-optical
wavelengths ($\lambda\lesssim4500$ \AA) (i) enables the recovery of 2-4 times
as many elements as red-optical spectroscopy
($5000\lesssim\lambda\lesssim10000$ \AA) at similar or higher resolutions
($R\sim 10000$) and (ii) can constrain the abundances of several neutron
capture elements to $\lesssim$0.3 dex. We further show that high-resolution
($R\gtrsim 20000$), low S/N ($\sim$10 pixel$^{-1}$) spectra contain rich
abundance information when modeled with full spectral fitting techniques. We
demonstrate that JWST/NIRSpec and ELTs can recover (i) $\sim$10 and 30
elements, respectively, for metal-poor red giants throughout the Local Group
and (ii) [Fe/H] and [$\alpha$/Fe] for resolved stars in galaxies out to several
Mpc with modest integration times. We show that select literature abundances
are within a factor of $\sim$2 (or better) of our CRLBs. We suggest that, like
ETCs, CRLBs should be used when planning stellar spectroscopic observations. We
include an open source python package, \texttt{Chem-I-Calc}, that allows users
to compute CRLBs for spectrographs of their choosing.
