  Now that ALMA is reaching its full capabilities, observations of sub-mm
emission line deep fields become feasible. We couple a semi-analytic model of
galaxy formation with a radiative transfer code to make predictions for the
luminosity function of CO J=1-0 up to CO J=6-5 and [CII] at redshifts z=0-6. We
find that: 1) our model correctly reproduces the CO and [CII] emission of low-
and high-redshift galaxies and reproduces the available constraints on the CO
luminosity function at z<2.75; 2) we find that the CO and [CII] luminosity
functions of galaxies increase from z = 6 to z = 4, remain relatively constant
till z = 1 and rapidly decrease towards z = 0. The galaxies that are brightest
in CO and [CII] are found at z~2; 3) the CO J=3-2 emission line is most
favourable to study the CO luminosity and global H2 mass content of galaxies,
because of its brightness and observability with currently available sub-mm and
radio instruments; 4) the luminosity functions of high-J CO lines show stronger
evolution than the luminosity functions of low-J CO lines; 5) our model barely
reproduces the available constraints on the CO and [CII] luminosity function of
galaxies at z>1.5 and the CO luminosity of individual galaxies at intermediate
redshifts. We argue that this is driven by a lack of cold gas in galaxies at
intermediate redshifts as predicted by cosmological simulations of galaxy
formation.
