The art of modelling CO, [C I], and [C II] in cosmological galaxy formation models

The advent of new sub-millimetre (sub-mm) observational facilities has stimulated the desire to model the sub-mm line emission of galaxies within cosmological galaxy formation models. This is typically done by applying sub-resolution recipes to describe the properties of the unresolved interstellar medium (ISM). While there is freedom in how one implements sub-resolution recipes, the impact of various choices has yet to be systematically explored. We combine a semi-analytic model of galaxy formation with chemical equilibrium networks and numerical radiative transfer models and explore how different choices for the sub-resolution modelling affect the predicted CO, [C I], and [C II] emission of galaxies. A key component for a successful model includes a molecular cloud mass-size relation and scaling for the ultraviolet and cosmic ray radiation field that depend on local ISM properties. Our most successful model adopts a Plummer radial density profile for gas within molecular clouds. Different assumptions for the clumping of gas within molecular clouds and changes in the molecular cloud mass distribution function hardly affect the CO, [C I], and [C II] luminosities of galaxies. At fixed star formation rate, the [C II]-SFR ratio of galaxies scales inversely with the pressure acting on molecular clouds, increasing the molecular clouds density and hence decreasing the importance of [C II] line cooling. We find that it is essential that a wide range of sub-mm emission lines arising in vastly different phases of the ISM are used as model constraints in order to limit the freedom in sub-grid choices.