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.
