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 $\scriptstyle \rm II$ ] 158 µm 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 µm. This method can be applied to all Atacama Large Millimeter Array (ALMA) and NOEMA [C $\scriptstyle \rm II$ ] observations, and exploited in ALMA Large Programs such as ALPINE and REBELS targeting [C $\scriptstyle \rm II$ ] emitters at high-z. We also provide a physical interpretation of the empirical relation recently found between molecular gas mass and [C $\scriptstyle \rm II$ ] luminosity. We derive an analogous relation linking the total gas surface density and [C $\scriptstyle \rm II$ ] surface brightness. By combining the two, we predict the cosmic evolution of the surface density ratio $\Sigma _{\rm H_2} / \Sigma _{\rm gas}$ . We find that $\Sigma _{\rm H_2} / \Sigma _{\rm gas}$ slowly increases with redshift, which is compatible with current observations at 0 < z < 4.