The [C II] line emission as an interstellar medium probe in the MARIGOLD galaxies

The [C II] fine-structure line at 157.74 $\mu$m is one of the brightest far-infrared emission lines and an important probe of galaxy properties like the star formation rate (SFR) and the molecular gas mass ($M_{\mathrm{mol}}$). Using high-resolution numerical simulations, we test the reliability of the [C II] line as a tracer of $M_{\mathrm{mol}}$ in high-redshift galaxies and investigate secondary dependences of the [C II]-$M_{\mathrm{mol}}$ relation on the SFR and metallicity. We investigate the time evolution of the [C II] luminosity function (LF) and the relative spatial extent of [C II] emission and star formation. We post-process galaxies from the MARIGOLD simulations at redshifts $3 \le z \leq 7$ to obtain their [C II] emission. These simulations were performed with the sub-grid chemistry model, HYACINTH, to track the non-equilibrium abundances of $\mathrm{H_2}$, $\mathrm{CO}$, $\rm C$ and $\mathrm{C^+}$ on the fly. Based on a statistical sample of galaxies at these redshifts, we investigate correlations between the [C II] line luminosity, L([C II]), and the SFR, the $M_{\mathrm{mol}}$, the total gas mass and the metal mass in gas phase ($M_{\mathrm{metal}}$). We find that accounting for secondary dependencies in the L([C II])-$M_{\mathrm{mol}}$ relation improves the $M_{\mathrm{mol}}$ prediction by a factor of 2.3. The [C II] emission in our simulated galaxies shows the tightest correlation with $M_{\mathrm{metal}}$. About 20% (10%) of our simulated galaxies at $z=5$ ($z=4$) have [C II] emission extending $\geq 2$ times farther than the star formation activity. The [C II] LF evolves rapidly and is always well approximated by a double power law that does not show an exponential cutoff at the bright end. We record a 600-fold increase in the number density of L([C II]) $\sim 10^9 \, \mathrm{L_{\odot}}$ emitters in 1.4 Gyr.