Deriving a multivariate CO-to-H$_2$ conversion function using the
  [CII]/CO(1-0) ratio and its application to molecular gas scaling relations
  We present Herschel PACS observations of the [CII] 158 micron emission line
in a sample of 24 intermediate mass (9<logM$_\ast$/M$_\odot$<10) and low
metallicity (0.4< Z/Z$_\odot$<1.0) galaxies from the xCOLD GASS survey.
Combining them with IRAM CO(1-0) measurements, we establish scaling relations
between integrated and molecular region [CII]/CO(1-0) luminosity ratios as a
function of integrated galaxy properties. A Bayesian analysis reveals that only
two parameters, metallicity and offset from the star formation main sequence,
$\Delta$MS, are needed to quantify variations in the luminosity ratio;
metallicity describes the total dust content available to shield CO from UV
radiation, while $\Delta$MS describes the strength of this radiation field. We
connect the [CII]/CO luminosity ratio to the CO-to-H$_2$ conversion factor and
find a multivariate conversion function $\alpha_{CO}$, which can be used up to
z~2.5. This function depends primarily on metallicity, with a second order
dependence on $\Delta$MS. We apply this to the full xCOLD GASS and PHIBSS1
surveys and investigate molecular gas scaling relations. We find a flattening
of the relation between gas mass fraction and stellar mass at
logM$_\ast$/M$_\odot$<10. While the molecular gas depletion time varies with
sSFR, it is mostly independent of mass, indicating that the low L$_{CO}$/SFR
ratios long observed in low mass galaxies are entirely due to photodissociation
of CO, and not to an enhanced star formation efficiency.
