A novel approach to investigate chemical inhomogeneities in GRB host galaxies: The $Z_{\rm abs} - Z_{\rm emiss}$ relation.

Models of chemical enrichment and inhomogeneity in high-redshift galaxies are challenging to constrain observationally. In this work, we discuss a novel approach to probe chemical inhomogeneities within long Gamma-Ray Burst (GRB) host galaxies, by comparing the absorption metallicity, Z_abs, from the GRB afterglow (which probes the environment along the line of sight) with the emission-line metallicity, Z_emiss, measured via slit spectroscopy. Using the IllustrisTNG simulation, the theoretical relationship between these metallicity metrics is explored for a range of GRB formation models, varying the GRB progenitor metallicity threshold. For galaxies with fixed Z_emiss, the median value of Z_abs depends strongly on the GRB progenitor threshold metallicity, with Z_abs significantly lower than Z_emiss for high metallicity hosts. Conversely, at fixed Z_abs, the median value of Z_emiss depends primarily on the metallicity distribution of galaxies in IllustrisTNG and their chemical inhomogeneities, offering a GRB-model-independent way to constrain these processes observationally. Currently, only one host galaxy has data for both absorption and emission metallicities (GRB121014A). We re-analyse the emission spectrum and compare the inferred metallicity Z_emiss to a recent Bayesian determination of Z_abs, finding $\log(Z_{\rm emiss}/Z_{\odot}) = \log(Z_{\rm abs}/Z_{\odot}) +0.35^{+ 0.14}_{- 0.25}$, within ~2 standard deviations of predictions from the IllustrisTNG simulation. Future observations with the James Webb Space Telescope will be able to measure Z_emiss for 4 other GRB hosts with known Z_abs values, using ~2 hour observations. While small, the sample will provide preliminary constraints on the Z_abs-Z_emiss relation to test chemical enrichment schemes in cosmological simulations.