  The advent of new sub-millimeter observational facilities has stimulated the
desire to model the sub-mm line emission of galaxies within cosmological galaxy
formation models. This is typically done in post-processing by applying
sub-resolution recipes to describe the properties of the unresolved
interstellar medium. At the same time, while there is freedom in how one
implements these sorts of recipes, the impact of various choices has yet to be
systematically explored in simulations. In this paper, we do just that. 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-grid modeling affect the predicted CO, [CI], and [CII]
emission of galaxies. We find that 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 varies based on local ISM properties. Our
most successful model adopts a Plummer profile for the radial density
distribution of gas within molecular clouds. On the other hand, different
assumptions for the clumping of gas within molecular clouds and changes in the
slope of the molecular cloud mass distribution function hardly affect the CO,
[CI], and [CII] luminosities of galaxies. At fixed star-formation rate the
[CII]-SFR ratio of galaxies scales inversely with the pressure acting on
molecular clouds, increasing the molecular clouds density and hence decreasing
the importance of [CII] line cooling. Overall 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 and we present the details of the most successful model variant.
