The Evolution of Molecular Gas Fraction Traced by the CO Tully-Fisher Relation

Carbon monoxide (CO) observations show a luminosity$-$line-width correlation that evolves with redshift. We present a method to use CO measurements alone to infer the molecular gas fraction ($f_{\rm mol}$) and constrain the CO$-$H$_2$ conversion factor ($\alpha_{\rm CO}$). We compile from the literature spatially integrated low-$J$ CO observations of six galaxy populations, including a total of 449 galaxies between $0.01 \leq z \leq 3.26$. The CO data of each population provide an estimate of the $\alpha_{\rm CO}$-normalized mean molecular gas fraction ($f_{\rm mol}/\alpha_{\rm CO}$). The redshift evolution of the luminosity$-$line-width correlation thus indicates an evolution of $f_{\rm mol}/\alpha_{\rm CO}$. We use a Bayesian-based Monte-Carlo Markov Chain sampler to derive the posterior probability distribution functions of $f_{\rm mol}/\alpha_{\rm CO}$ for these galaxy populations, accounting for random inclination angles and measurement errors in the likelihood function. We find that the molecular gas fraction evolves rapidly with redshift, $f_{\rm mol} \propto (1+z)^\beta$ with $\beta \simeq 2$, for both normal star-forming and starburst galaxies. Furthermore, the evolution trend agrees well with that inferred from the Kennicutt-Schmidt relation and the star-forming main sequence. Finally, at $z < 0.1$ normal star-forming galaxies require a $\sim5\times$ larger $\alpha_{\rm CO}$ than starburst galaxies to match their molecular gas fractions, but at $z > 1$ both star-forming types exhibit sub-Galactic $\alpha_{\rm CO}$ values and normal star-forming galaxies appear more gas-rich than starbursts. Future applications of this method include calibrating Tully-Fisher relations without inclination correction and inferring the evolution of the atomic gas fraction with HI observations.