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Carbon monoxide (CO) observations show a luminosity$-$line-width correlationthat evolves with redshift. We present a method to use CO measurements alone toinfer the molecular gas fraction ($f_{\rm mol}$) and constrain the CO$-$H$_2$conversion factor ($\alpha_{\rm CO}$). We compile from the literature spatiallyintegrated low-$J$ CO observations of six galaxy populations, including a totalof 449 galaxies between $0.01 \leq z \leq 3.26$. The CO data of each populationprovide an estimate of the $\alpha_{\rm CO}$-normalized mean molecular gasfraction ($f_{\rm mol}/\alpha_{\rm CO}$). The redshift evolution of theluminosity$-$line-width correlation thus indicates an evolution of $f_{\rmmol}/\alpha_{\rm CO}$. We use a Bayesian-based Monte-Carlo Markov Chain samplerto derive the posterior probability distribution functions of $f_{\rmmol}/\alpha_{\rm CO}$ for these galaxy populations, accounting for randominclination angles and measurement errors in the likelihood function. We findthat the molecular gas fraction evolves rapidly with redshift, $f_{\rm mol}\propto (1+z)^\beta$ with $\beta \simeq 2$, for both normal star-forming andstarburst galaxies. Furthermore, the evolution trend agrees well with thatinferred from the Kennicutt-Schmidt relation and the star-forming mainsequence. Finally, at $z < 0.1$ normal star-forming galaxies require a$\sim5\times$ larger $\alpha_{\rm CO}$ than starburst galaxies to match theirmolecular gas fractions, but at $z > 1$ both star-forming types exhibitsub-Galactic $\alpha_{\rm CO}$ values and normal star-forming galaxies appearmore gas-rich than starbursts. Future applications of this method includecalibrating Tully-Fisher relations without inclination correction and inferringthe evolution of the atomic gas fraction with HI observations.

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