Dust temperature in {ALMA} [C~ii]-detected high-z galaxies

At redshift z > 5, the far-infrared (FIR) continuum spectra of main-sequence galaxies are sparsely sampled, often with a single data point. The dust temperature T-d(,SED), thus has to be assumed in the FIR continuum fitting. This introduces large uncertainties regarding the derived dust mass (M-d), FIR luminosity, and obscured fraction of the star formation rate. These are crucial quantities to quantify the effect of dust obscuration in high-z galaxies. To overcome observation limitations, we introduce a new method that combines dust continuum information with the overlying [C II] 158 mu m line emission. By breaking the M-d T-d(,SED) degeneracy, with our method, we can reliably constrain the dust temperature with a single observation at 158 mu m. This method can be applied to all Atacama Large Millimeter Array (ALMA) and NOEMA [C II] observations, and exploited in ALMA Large Programs such as ALPINE and REBELS targeting [C II] emitters at high-z. We also provide a physical interpretation of the empirical relation recently found between molecular gas mass and [C II] luminosity. We derive an analogous relation linking the total gas surface density and [C II] surface brightness. By combining the two, we predict the cosmic evolution of the surface density ratio Sigma(H2) / Sigma(gas). We find that Sigma(H2)/ Sigma(gas) slowly increases with redshift, which is compatible with current observations at 0 < z < 4.