tokens: ∼10% at z = 18 to 23% at z = 12. While this is not an observable, it is an additional independent cross-check for our approach. 4.3. When Is <italic>dZ</italic> Important? In Figure 10 , we show the distribution of dZ as a function of physical parameters. Since dZ is most important in external enrichment, these panels indicate under what conditions external enrichment becomes important. For some stars with low [Fe/H] (≲−4), dZ is largely negative. This means that the metallicities of these stars would have been much higher if we assumed homogeneity. The clear cut at [Fe/H] ∼ −5.1 comes from our Population II star formation criterion. For stars with [Fe/H] > −5.1, we do not require carbon enhancement to form stars, whereas if [Fe/H] −5.1, we do require carbon enhancement (see Equation ( 2 )). Highly negative dZ means high (naively calculated) [Fe/H], which is therefore less likely to be carbon-enhanced and allowed to form stars. Figure 10. Distribution of dZ as a function of various physical properties of stars. (Top) For stars with [Fe/H] −4.5, external enrichment is the dominant channel. Therefore, inhomogeneity is important for such ultra-metal-poor stars. (Middle) For large, matured halos (), star formation has already begun; therefore, they are internally enriched. External enrichment is important when we consider star formation in small galaxies (). (Bottom) For star formation events at a very high redshift, external enrichment plays a role. The middle panel shows that in small halos, dZ can be largely negative. In massive galaxies (), star formation has already begun; therefore, they are internally enriched and well mixed. Combining with our former analysis of the dZ distribution, we find that the metallicity inhomogeneity is less significant in massive galaxies. The bottom panel shows that dZ is only important at high (although not the highest) redshifts. Low-redshift galaxies are homogeneous for the same reason as the massive galaxies. The highest-redshift galaxies () are homogeneous because they do not have enough time after the first star formation events to experience external enrichment. In summary, inhomogeneous metal mixing is important in low-mass halos at high redshift that are about to form EMP stars. 5. Discussion We have performed the first cosmologically representative analysis of metal mixing in high-redshift galaxies. We derived a physically motivated estimate of dZ , the metallicity difference between star-forming and all gas, and find that the distribution of dZ is very different between halos with stars and halos without stars. This bimodality can be understood by assuming that two processes are at work: internal enrichment and external enrichment. Halos without stars have not experienced any star formation events, so they can be identified as externally enriched halos. In external enrichment, the momentum of metal-rich wind is not strong. If a galaxy already has a dense gas cloud when the external enrichment takes
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