First joint absorption and T$_e$-based metallicity measured in a GRB host galaxy at $z=4.28$ using JWST/NIRSpec

We present the first gamma-ray burst (GRB) host galaxy with a measured absorption line and electron temperature (T$_e$) based metallicity, using the temperature sensitive [OIII]$\lambda$4363 auroral line detected in the JWST/NIRSpec spectrum of the host of GRB 050505 at redshift $z=4.28$. We find that the metallicity of the cold interstellar gas, derived from the absorption lines in the GRB afterglow, of 12 + log(O/H)$\sim 7.7$ is in reasonable agreement with the temperature-based emission line metallicity in the warm gas of the GRB host galaxy, which has values of 12 + log(O/H) = 7.80$\pm$0.19 and 7.96$\pm$0.21 for two common indicators. When using strong emission line diagnostics appropriate for high-z galaxies and sensitive to ionisation parameter, we find good agreement between the strong emission line metallicity and the other two methods. Our results imply that, for the host of GRB050505, mixing between the warm and the cold ISM along the line of sight to the GRB is efficient, and that GRB afterglow absorption lines can be a reliable tracer of the metallicity of the galaxy. If confirmed with a large sample, this suggest that metallicities determined via GRB afterglow spectroscopy can be used to trace cosmic chemical evolution to the earliest cosmic epochs and in galaxies far too faint for emission line spectroscopy, even for JWST.