Filter Design for Estimation of Stellar Metallicity: Insights from Experiments with Gaia XP Spectra

We search for an optimal filter design for the estimation of stellar metallicity, based on synthetic photometry from Gaia XP spectra convolved with a series of filter-transmission curves defined by different central wavelengths and bandwidths. Unlike previous designs based solely on maximizing metallicity sensitivity, we find that the optimal solution provides a balance between the sensitivity and uncertainty of the spectra. With this optimal filter design, the best precision of metallicity estimates for relatively bright ($G \sim 11.5$) stars is excellent, $\sigma_{\rm [Fe/H]} = 0.034$\,dex for FGK dwarf stars, superior to that obtained utilizing custom sensitivity-optimized filters (e.g., SkyMapper\,$v$). By selecting hundreds of high-probabability member stars of the open cluster M67, our analysis reveals that the intrinsic photometric-metallicity scatter of these cluster members is only 0.036\,dex, consistent with this level of precision. Our results clearly demonstrate that the internal precision of photometric-metallicity estimates can be extremely high, even providing the opportunity to perform chemical tagging for very large numbers of field stars in the Milky Way. This experiment shows that it is crucial to take into account uncertainty alongside the sensitivity when designing filters for measuring the stellar metallicity and other parameters.