Metallicity fluctuation statistics in the interstellar medium and young stars - I. Variance and correlation

The distributions of a galaxy's gas and stars in chemical space encode a tremendous amount of information about that galaxy's physical properties and assembly history. However, present methods for extracting information from chemical distributions are based either on coarse averages measured over galactic scales (e.g. metallicity gradients) or on searching for clusters in chemical space that can be identified with individual star clusters or gas clouds on ~1 pc scales. These approaches discard most of the information, because in galaxies gas and young stars are observed to be distributed fractally, with correlations on all scales, and the same is likely to be true of metals. In this paper we introduce a first theoretical model, based on stochastically forced diffusion, capable of predicting the multiscale statistics of metal fields. We derive the variance, correlation function, and power spectrum of the metal distribution from first principles, and determine how these quantities depend on elements' astrophysical origin sites and on the large-scale properties of galaxies. Among other results, we explain for the first time why the typical abundance scatter observed in the interstellar media of nearby galaxies is ≈0.1 dex, and we predict that this scatter will be correlated on spatial scales of ~0.5-1 kpc, and over time-scales of~100-300 Myr. We discuss the implications of our results for future chemical tagging studies.