We use the first systematic data sets of CO molecular line emission in z ∼ 1-3 normal star-forming galaxies (SFGs) for a comparison of the dependence of galaxy-averaged star formation rates on molecular gas masses at low and high redshifts, and in different galactic environments. Although the current high-z samples are still small and biased towards the luminous and massive tail of the actively star-forming 'main-sequence', a fairly clear picture is emerging. Independent of whether galaxy-integrated quantities or surface densities are considered, low-and high-z SFG populations appear to follow similar molecular gas-star formation relations with slopes 1.1 to 1.2, over three orders of magnitude in gas mass or surface density. The gas-depletion time-scale in these SFGs grows from 0.5 Gyr at z ∼ 2 to 1.5 Gyr at z ∼ 0. The average corresponds to a fairly low star formation efficiency of 2 per cent per dynamical time. Because star formation depletion times are significantly smaller than the Hubble time at all redshifts sampled, star formation rates and gas fractions are set by the balance between gas accretion from the halo and stellar feedback.In contrast, very luminous and ultraluminous, gas-rich major mergers at both low and high z produce on average four to 10 times more far-infrared luminosity per unit gas mass. We show that only some fraction of this difference can be explained by uncertainties in gas mass or luminosity estimators; much of it must be intrinsic. A possible explanation is a top-heavy stellar mass function in the merging systems but the most likely interpretation is that the star formation relation is driven by global dynamical effects. For a given mass, the more compact merger systems produce stars more rapidly because their gas clouds are more compressed with shorter dynamical times, so that they churn more quickly through the available gas reservoir than the typical normal disc galaxies. When the dependence on galactic dynamical Based on observations with the Plateau de Bure millimetre interferometre, operated by the Institute for Radio Astronomy in the Millimetre Range (IRAM), which is funded by a partnership of INSU/CNRS (France), MPG (Germany) and IGN (Spain).

This paper provides an update of our previous scaling relations (Genzel et al. 2015) between galaxy integrated molecular gas masses, stellar masses and star formation rates, in the framework of the star formation main-sequence (MS), with the main goal to test for possible systematic effects. For this purpose our new study combines three independent methods of determining molecular gas masses from CO line fluxes, far-infrared dust spectral energy distributions, and ~1mm dust photometry, in a large sample of 1444 star forming galaxies (SFGs) between z=0 and 4. The sample covers the stellar mass range log(M*/M)=9.0-11.8, and star formation rates relative to that on the MS, δMS=SFR/SFR(MS), from 10 -1.3 to 10 2.2 . Our most important finding is that all data sets, despite the different techniques and analysis methods used, follow the same scaling trends, once method-to-method zero point offsets are minimized and uncertainties are properly taken into account. The molecular gas depletion time tdepl, defined as the ratio of molecular gas mass to star formation rate, scales as (1+z) -0.6 × (δMS) -0.44 , and is only weakly dependent on stellar mass. The ratio of molecular-to-stellar mass μgas depends on (1+z) 2.5 × (δMS) 0.52 × (M*) -0.36 , which tracks the evolution of the specific star formation rate. The redshift dependence of μgas requires a curvature term, as may the mass-dependences of tdepl and μgas. We find no or only weak correlations of tdepl and μgas with optical size R or surface density once one removes the above scalings, but we caution that optical sizes may not be appropriate for the high gas and dust columns at high-z.