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The evolution of the galaxy stellar mass–star formation rate relationship ( M  * –SFR) provides key constraints on the stellar mass assembly histories of galaxies. For star‐forming galaxies,  M  * –SFR is observed to be fairly tight with a slope close to unity from  z ∼ 0 → 2, and it evolves downwards roughly independently of  M  * . Simulations of galaxy formation reproduce these trends, broadly independent of modelling details, owing to the generic dominance of smooth and steady cold accretion in these systems. In contrast, the observed amplitude of the  M  * –SFR relation evolves markedly differently than in models, indicating either that stellar mass assembly is poorly understood or that observations have been misinterpreted. Stated in terms of a star formation activity parameter  α sf ≡ ( M  * /SFR)/( t  Hubble –1 Gyr) , models predict a constant α sf ∼ 1 out to redshifts  z ∼ 4+, while the observed  M  * –SFR relation indicates that α sf  increases by approximately three times from  z ∼ 2 until today. The low α sf  (i.e. rapid star formation) at high  z  not only conflicts with models, but also difficult to reconcile with other observations of high‐ z  galaxies, such as the small scatter in  M  * –SFR, the slow evolution of star‐forming galaxies at  z ∼ 2–4 and the modest passive fractions in mass‐selected samples. Systematic biases could significantly affect measurements of  M  *  and SFR, but detailed considerations suggest that none are obvious candidates to reconcile the discrepancy. A speculative solution is considered in which the stellar initial mass function (IMF) evolves towards more high‐mass star formation at earlier epochs. Following Larson, a model is investigated in which the characteristic mass  where the IMF turns over increases with redshift. Population synthesis models are used to show that the observed and predicted  M  * –SFR evolution may be brought into broad agreement if  out to  z ∼ 2. Such IMF evolution matches recent observations of cosmic stellar mass growth, and the resulting   z = 0  cumulative IMF is similar to the ‘paunchy’ IMF favoured by Fardal et al. to reconcile the observed cosmic star formation history with present‐day fossil light measures.

The galaxy stellar mass–star formation rate relation: evidence for an evolving stellar initial mass function?