  Deep Herschel imaging and 12CO(2-1) line luminosities from the IRAM PdBI are
combined for a sample of 17 galaxies at z>1 from the GOODS-N field. The sample
includes galaxies both on and above the main sequence (MS) traced by
star-forming galaxies in the SFR-M* plane. The far-infrared data are used to
derive dust masses, Mdust. Combined with an empirical prescription for the
dependence of the gas-to-dust ratio on metallicity (GDR), the CO luminosities
and Mdust values are used to derive for each galaxy the CO-to-H2 conversion
factor, alpha_co. Like in the local Universe, the value of alpha_co is a factor
of ~5 smaller in starbursts compared to normal star-forming galaxies (SFGs). We
also uncover a relation between alpha_co and dust temperature (Tdust; alpha_co
decreasing with increasing Tdust) as obtained from modified blackbody fits to
the far-infrared data. While the absolute normalization of the alpha_co(Tdust)
relation is uncertain, the global trend is robust against possible systematic
biases in the determination of Mdust, GDR or metallicity. Although we cannot
formally distinguish between a step and a smooth evolution of alpha_co with the
dust temperature, we can conclude that in galaxies of near-solar metallicity, a
critical value of Tdust=30K can be used to determine whether the appropriate
alpha_co is closer to the starburst value (1.0 Msun(K kms pc^2)^-1, if
Tdust>30K) or closer to the Galactic value (4.35 Msun (K kms pc^2)^-1, if
Tdust<30K). This indicator has the great advantage of being less subjective
than visual morphological classifications of mergers/SFGs, which can be
difficult at high z because of the clumpy nature of SFGs. In the absence of
far-infrared data, the offset of a galaxy from the main sequence (i.e.,
log[SSFR(galaxy)/SSFR_MS(M*,z)]) can be used to identify galaxies requiring the
use of an alpha_co conversion factor lower than the Galactic value.
