Metallicity dependence of the CO-to-H$_2$ and the [CI]-to-H$_2$ conversion factors in galaxies

Understanding the molecular gas content in the interstellar medium (ISM) is crucial for studying star formation and galaxy evolution. The CO-to-H$_2$ ($X_{\rm CO}$) and the [CI]-to-H$_2$ ($X_{\rm CI}$) conversion factors are widely used to estimate the molecular mass content in galaxies. However, these factors depend on many ISM environmental parameters. This work investigates the dependence of $X_{\rm CO}$ and $X_{\rm CI}$ on these parameters, with a focus on the low-metallicity $\alpha$-enhanced ISM ($\rm [C/O]<0$), to provide improved tracers of molecular gas in diverse conditions. We used the PDFchem algorithm, coupled with a database of 3D-PDR models. These account for a wide range of metallicities, dust-to-gas mass ratios, FUV intensities, and cosmic-ray ionization rates. The conversion factors were computed by integrating the PDR properties over log-normal column density distributions ($A_{\rm V}$-PDFs) representing various cloud types. The $X_{\rm CO}$ factor increases significantly with decreasing metallicity, exceeding $\sim\!\!1000$ times the Galactic value at ${\rm [O/H] = -1.0}$ under $\alpha$-enhanced conditions, as opposed to $\sim\!\!300$ times under non-$\alpha$-enhanced conditions (${\rm [C/O]=0}$). In contrast, $X_{\rm CI}$ shows a more gradual variation with metallicity, making it a more reliable tracer of molecular gas in metal-poor environments under most conditions. The fraction of `CO-dark' molecular gas increases dramatically in low-metallicity regions, exceeding 90\% at ${\rm [O/H] = -1.0}$, particularly in diffuse clouds and environments with strong FUV radiation fields. We recommend the use of the $\log_{10}X_{\rm CO}\simeq-2.41Z+41.3$ relation for the CO-to-H$_2$ conversion factor, and the $\log_{10}X_{\rm CI}\simeq-0.99Z+29.7$ relation for the [CI]-to-H$_2$, where $Z=12+\log_{10}({\rm O/H})$.