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.