$\rm [C_{II}]$ 158 $\rm μm$ 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 ($L_{\rm [C_{II}]}$) and star formation rate (SFR), suggesting that $L_{\rm [C_{II}]}$ may be a useful SFR tracer for galaxies. Some other galaxy populations, however, are found to have lower $L_{\rm [C_{II}]}{}/{}\rm SFR$ than the local SFGs, including the infrared-luminous, starburst galaxies at low and high redshifts, as well as some moderately star-forming galaxies at the epoch of re-ionization (EoR). The origin of this `$\rm [C_{II}]$ deficit' is unclear. In this work, we study the $L_{\rm [C_{II}]}$-SFR relation of galaxies using a sample of $z=0-8$ galaxies with $M_*\approx10^7-5\times10^{11}\,M_\odot$ extracted from cosmological volume and zoom-in simulations from the Feedback in Realistic Environments (FIRE) project. We find a simple analytic expression for $L_{\rm [C_{II}]}$/SFR of galaxies in terms of the following parameters: mass fraction of $\rm [C_{II}]$-emitting gas ($f_{\rm [C_{II}]}$), gas metallicity ($Z_{\rm gas}$), gas density ($n_{\rm gas}$) and gas depletion time ($t_{\rm dep}{}={}M_{\rm gas}{}/{}\rm SFR$). We find two distinct physical regimes, where $t_{\rm dep}$ ($Z_{\rm gas}$) is the main driver of the $\rm [C_{II}]$ deficit in $\rm H_2$-rich ($\rm H_2$-poor) galaxies. The observed $\rm [C_{II}]$ 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 $\rm [C_{II}]$ deficit is a common phenomenon of galaxies, and caution needs to be taken when applying a constant $L_{\rm [C_{II}]}$-to-SFR conversion factor derived from local SFGs to estimate cosmic SFR density at high redshifts and interpret data from upcoming $\rm [C_{II}]$ line intensity mapping experiments.