[C II] 158 μm emission as an indicator of galaxy star formation rate

Observations of local star-forming galaxies (SFGs) show a tight correlation between their singly ionized carbon line luminosity ($L_{\rm [C\, {\small II}]}$) and star formation rate (SFR), suggesting that $L_{\rm [C\, {\small II}]}$ may be a useful SFR tracer for galaxies. Some other galaxy populations, however, are found to have lower $L_{\rm [C\, {\small II}]}{}/{}\rm SFR$ than local SFGs, including the infrared (IR)-luminous, starburst galaxies at low and high redshifts as well as some moderately SFGs at the epoch of re-ionization (EoR). The origins of this '$\rm [C\, {\small II}]$ deficit' is unclear. In this work, we study the $L_{\rm [C\, {\small II}]}$-SFR relation of galaxies using a sample of z = 0-8 galaxies with $M_*\approx 10^7-5\times 10^{11}\, \mathrm{M}_\odot$ extracted from cosmological volume and zoom-in simulations from the Feedback in Realistic Environments (FIRE) project. We find a simple analytic expression for $L_{\rm [C\, {\small II}]}$/SFR of galaxies in terms of the following parameters: mass fraction of $\rm [C\, {\small II}]$-emitting gas ($f_{\rm [C\, {\small II}]}$), gas metallicity (Z_gas), gas density (n_gas), and gas depletion time ($t_{\rm dep}{}={}M_{\rm gas}{}/{}\rm SFR$). We find two distinct physical regimes: $\rm H_2$-rich galaxies, where t_dep is the main driver of the $\rm [C\, {\small II}]$ deficit and $\rm H_2$-poor galaxies where Z_gas is the main driver. The observed $\rm [C\, {\small II}]$ deficit of IR-luminous galaxies and early EoR galaxies, corresponding to the two different regimes, is due to short gas depletion time and low gas metallicity, respectively. Our result indicates that the $\rm [C\, {\small II}]$ deficit is a common phenomenon of galaxies, and caution needs to be taken when applying a constant $L_{\rm [C\, {\small II}]}$-to-SFR conversion factor derived from local SFGs to estimate cosmic SFR density at high redshifts and interpret data from upcoming $\rm [C\, {\small II}]$ line intensity mapping experiments.