Effects of dust evolution on the abundances of CO and H$_2$

The CO-to-H$_2$ conversion factor ($X_\mathrm{CO}$) is known to correlate with the metallicity ($Z$). The dust abundance, which is related to the metallicity, is responsible for this correlation through dust shielding of dissociating photons and H$_2$ formation on dust surfaces. In this paper, we investigate how the relation between dust-to-gas ratio and metallicity ($\mathcal{D}$--$Z$ relation) affects the H$_2$ and CO abundances (and $X_\mathrm{CO}$) of a `molecular' cloud. For the $\mathcal{D}$--$Z$ relation, we adopt a dust evolution model developed in our previous work, which treats the evolution of not only dust abundance but also grain sizes in a galaxy. Shielding of dissociating photons and H$_2$ formation on dust are solved consistently with the dust abundance and grain sizes. As a consequence, our models {predict consistent metallicity dependence of $X_\mathrm{CO}$ with observational data}. Among various processes driving dust evolution, grain growth by accretion has the largest impact on the $X_\mathrm{CO}$--$Z$ relation. The other processes also have some impacts on the $X_\mathrm{CO}$--$Z$ relation, but their effects are minor compared with the scatter of the observational data at the metallicity range ($Z\gtrsim 0.1$ Z$_\odot$) where CO could be detected. We also find that dust condensation in stellar ejecta has a dramatic impact on the H$_2$ abundance at low metallicities ($\lesssim 0.1$ Z$_\odot$), relevant for damped Lyman $\alpha$ systems and nearby dwarf galaxies, and that the grain size dependence of H$_2$ formation rate is also important.