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Modelling the molecular gas that is routinely detected through CO observations of high-redshift galaxies constitutes a major challenge for ab initio simulations of galaxy formation. We carry out a suite of cosmological hydrodynamic simulations to compare three approximate methods that have been used in the literature to track the formation and evolution of the simplest and most abundant molecule, H2. Namely, we consider (i) a semi-empirical procedure that associates H2 to dark-matter haloes based on a series of scaling relations inferred from observations, (ii) a model that assumes chemical equilibrium between the H2 formation and destruction rates, and (iii) a model that fully solves the out-of-equilibrium rate equations and accounts for the unresolved structure of molecular clouds. We study the impact of finite spatial resolution and show that robust H2 masses at redshift z ≈ 4 can only be obtained for galaxies that are sufficiently metal enriched in which H2 formation is fast. This corresponds to H2 reservoirs with masses MH2 ≿ 6 × 109 M☉. In this range, equilibrium and non-equilibrium models predict similar molecular masses (but different galaxy morphologies) while the semi-empirical method produces less H2. The star formation rates as well as the stellar and H2 masses of the simulated galaxies are in line with those observed in actual galaxies at similar redshifts that are not massive starbursts. The H2 mass functions extracted from the simulations at z ≈ 4 agree well with recent observations that only sample the high-mass end. However, our results indicate that most molecular material at high z lies yet undetected in reservoirs with 109 < MH2 < 1010 M☉.
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