tokens: an ideal tool with which to calibrate other tracers of star formation. Finally, CO emission is the consequence of the interaction between CO molecules and molecular hydrogen (H 2 ) in regions where stars are formed – giant molecular clouds (GMCs). CO emission is correlated with the virial mass of GMCs observed in the Milky Way and in nearby spiral galaxies (Young Scoville 1991 ). Historically, the IR–RC–CO parameters have been studied using three empirical relationships, each one having different properties: first, the RC–IR correlation (e.g. Yun, Reddy Condon 2001 ; Ivison et al. 2010 ; Thomson et al. 2014 ; Magnelli et al. 2015 ; Dumas et al. 2011 ; Tabatabaei et al. 2013a , b ), valid up to z ∼ 2 and at a resolution of ≥ 500 pc. Second, the Schmidt–Kennicutt (SK) IR–CO law (Kennicutt 1998 ; Kennicutt Evans 2012 ), valid for bright galaxies with a high SFR as well as for low-surface-brightness galaxies with a low SFR. The form of the SK law is still debated because observations of disc and starburst galaxies have shown that this relationship can be fitted either by a single or by a bi-modal solution for the redshift range 0.05–2.5 (e.g. Genzel et al. 2010 ; Daddi et al. 2010 ; Ivison et al. 2011 ; Kennicutt Evans 2012 ; Tacconi et al. 2013 ; Santini et al. 2014 ; Freundlich et al. 2019 ). The slope of the SK law, which defines the depletion time of the gas in a galaxy (τ dep ≡ M gas /SFR), is central to our understanding of the mechanisms that govern star formation. According to models, a linear slope implies that star-formation activity is not driven only by the self-gravity of the galaxy (Semenov, Kravtsov Gnedin 2017 , 2019 ). Several studies dedicated to the spatially resolved KS relationship on subkiloparsec scales found a wide range in the value of the slope (indicated by the power-law index, N ∼ 0.6–3; ) of the KS relationship (Bigiel et al. 2008 ; Kennicutt Evans 2012 ; Rahman et al. 2011 ; Viaene et al. 2014 ; Casasola et al. 2015 ). The spread in the value of N may be intrinsic, suggesting that different star formation relationships exist; alternatively, it may result from the assumptions adopted in each study. Daddi et al. ( 2010 ) suggest that L IR (tracing SFR) and (tracing molecular gas mass) show no evolution in normal star-forming galaxies up to z = 1.5. However, Tacconi et al. ( 2018 ) suggest that τ dep ∝ (1 + z ) −0.62 , implying that τ dep decreases by a factor of ~2×, up to the redshift limit explored in our study, z = 1.5. Third, the RC–CO correlation (e.g. Adler et al. 1991 ; Murgia et al. 2002 , 2005 ; Leroy et al. 2005 ; Paladino et al. 2006 ; Schinnerer et al. 2013 ; Liu et al. 2015 ) is valid for a variety of galaxy types and also down to scales of a few hundred parsecs in some selected local galaxies. In this work, we derived equation ( 3 ) by combining RC–IR and IR–CO relationships from the literature to obtain the molecular gas mass density of the Universe, after deriving the integrated MGMF from measurements of
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