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In this Chapter we have discussed a wide range of the physical processes that must be accounted for in the theoretical modeling of the first galaxies. We have made important distinctions between the formation of the first stars in minihalos and star formation in the atomic cooling halos hosting the first galaxies, highlighting how the cooling properties of the gas assembled into the first galaxies are altered by high energy radiation and by the injection of heavy elements from the first supernovae. While this has not been a complete review of the theory of the formation of the first galaxies, it has hopefully served to illustrate, from basic principles where possible, much of the physics that comes into play in their study. The reader is referred to the many excellent articles in the bibliography below for more in-depth study on the topic.

In closing, it is critical to point out that without accounting for all of the effects we have discussed together, one is left with an incomplete understanding of the first galaxies. For instance, as we have seen, the radiation from the first stars can ionize the gas and trigger HD cooling, but it can also easily destroy H2 and HD molecules. As well, while the first supernovae may enrich much of the gas from which the first galaxies form to a level above the critical metallicity needed for low-mass Pop II star formation, much of the dense gas in minihalos may not be efficiently mixed with the metal-enriched ejecta. Similarly, black holes may only form by direct collapse in rare regions in which the LW background radiation field is elevated, but the same stars which likely produce this radiation may also enrich the gas when they explode as supernovae, possibly precluding this mode of black hole formation. A complete and consistent picture of the formation of the first galaxies only emerges when accounting for star and black hole formation, metal enrichment, and radiative feedback all together in the full cosmological context.

Making this task especially daunting is the range of scales that must be taken into account. The gas clouds which collapse to form stars are on sub-parsec scales, metal-enrichment from the first supernovae occurs on parsec to kiloparsec scales, and the radiation emitted by the first stars can impact regions on kiloparsec or even megaparsec scales. Thus, simulations must ultimately resolve an enormous range of scales in order to capture all of the important physical processes that come into play. While we have introduced the results of numerous analytical calculations and simulations, none of them alone captures all of the processes we have discussed simultaneously. Indeed, this stands as one of the primary challenges to making detailed predictions of the nature of the first galaxies.

The author is grateful to the editors for the invitation to contribute this Chapter, as well as to Bhaskar Agarwal, Volker Bromm, Umberto Maio, and Eyal Neistein for helpful comments on an earlier draft of this work. Credit also goes to Chalence Safranek-Shrader for identifying an error (now corrected) in equation (29), as well as to Daisuke Nakauchi for identifying an inconsistency (now corrected) between equations (25) and (26).

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