  We predict the emission line luminosity functions (LFs) of the first 10
rotational transitions of CO in galaxies at redshift z=0 to z=10. This
prediction relies on a recently presented simulation of the molecular cold gas
content in ~3e7 evolving galaxies based on the Millennium Simulation. We
combine this simulation with a model for the conversion between molecular mass
and CO-line intensities, which incorporates the following mechanisms: (i)
molecular gas is heated by the CMB, starbursts (SBs), and active galactic
nuclei (AGNs); (ii) molecular clouds in dense or inclined galaxies can overlap;
(iii) compact gas can attain a smooth distribution in the densest part of
disks; (iv) CO-luminosities scale with metallicity changes between galaxies;
(v) CO-luminosities are always detected against the CMB. We analyze the
relative importance of these effects and predict the cosmic evolution of the
CO-LFs. The most notable conclusion is that the detection of regular galaxies
(i.e. no AGN, no massive SB) at high z>7 in CO-emission will be dramatically
hindered by the weak contrast against the CMB, in contradiction to earlier
claims that CMB-heating will ease the detection of high-redshift CO. The full
simulation of extragalactic CO-lines and the predicted CO-LFs at any redshift
can be accessed online, prior registration required} and they should be useful
for the modeling of CO-line surveys with future telescopes, such as ALMA, the
LMT, or the SKA.
