Tracing Molecular Gas in z $\simeq$ 6 Galaxies with [C${\rm \scriptsize II}$]

We investigate the fine-structure [C${\rm \scriptsize II}$] line at $158\,\mu$m as a molecular gas tracer by analyzing the relationship between molecular gas mass ($M_{\rm mol}$) and [C${\rm \scriptsize II}$] line luminosity ($L_{\rm [CII]}$) in 11,125 $z\simeq 6$ star-forming, main sequence galaxies from the SIMBA simulations, with line emission modeled by S\'IGAME. Though most ($\sim 50-100\,\%$) of the gas mass in our simulations is ionized, the bulk ($> 50\,\%$) of the [C${\rm \scriptsize II}$] emission comes from the molecular phase. We find a sub-linear (slope $0.78\pm 0.01$) $\log L_{\rm [CII]}-\log M_{\rm mol}$ relation, in contrast with the linear relation derived from observational samples of more massive, metal-rich galaxies at $z \lesssim 6$. We derive a median [C${\rm \scriptsize II}$]-to-$M_{\rm mol}$ conversion factor of $\alpha_{\rm [CII]} \simeq 18\,{\rm M_{\rm \odot}/L_{\rm \odot}}$. This is lower than the average value of $\simeq 30\,{\rm M_{\rm \odot}/L_{\rm \odot}}$ derived from observations, which we attribute to lower gas-phase metallicities in our simulations. Thus, a lower, luminosity-dependent, conversion factor must be applied when inferring molecular gas masses from [C${\rm \scriptsize II}$] observations of low-mass galaxies. For our simulations, [C${\rm \scriptsize II}$] is a better tracer of the molecular gas than CO $J=1-0$, especially at the lowest metallicities, where much of the gas is 'CO-dark'. We find that $L_{\rm [CII]}$ is more tightly correlated with $M_{\rm mol}$ than with star-formation rate (${\rm SFR}$), and both the $\log L_{\rm [CII]}-\log M_{\rm mol}$ and $\log L_{\rm [CII]}-\log {\rm SFR}$ relations arise from the Kennicutt-Schmidt relation. Our findings suggest that $L_{\rm [CII]}$ is a promising tracer of the molecular gas at the earliest cosmic epochs.