We devise a physical model of formation and distribution of molecular gas clouds in galaxies. We use the model to predict the intensities of rotational transition lines of carbon monoxide (CO) and the molecular hydrogen (H$_{rm 2}$) abundance. Using the outputs of Illustris-TNG cosmological simulations, we populate molecular gas clouds of unresolved sizes in individual simulated galaxies, where the effect of the interstellar radiation field with dust attenuation is also taken into account. We then use the publicly available code DESPOTIC to compute the CO line luminosities and H$_{rm 2}$ densities without assuming the CO-to-H$_{rm 2}$ conversion factor ($alpha_{rm CO}$). Our method allows us to study the spatial and kinematic structures traced by CO(1-0) and higher transition lines. We compare the CO luminosities and H$_{rm 2}$ masses with recent observations of galaxies at low and high redshifts. Our model reproduces well the observed CO-luminosity function and the estimated H$_{rm 2}$ mass in the local Universe. About ten per cent of molecules in the Universe reside in dwarf galaxies with stellar masses lower than $10^9~{rm M_odot}$, but the galaxies are generally `CO-dark and have typically high $alpha_{rm CO}$. Our model predicts generally lower CO line luminosities than observations at redshifts $zgtrsim 1$--$2$. We argue that the difference can be explained by the highly turbulent structure suggested for the high-redshift star-forming galaxies.
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