The evolution of the galaxy stellar mass--star formation rate relationship(M*-SFR) provides key constraints on the stellar mass assembly histories ofgalaxies. For star-forming galaxies, M*-SFR is observed to be fairly tight witha slope close to unity from z~0-2. Simulations of galaxy formation reproducethese trends owing to the generic dominance of smooth and steady cold accretionin these systems. In contrast, the amplitude of the M*-SFR relation evolvesmarkedly differently than in models. Stated in terms of a star formationactivity parameter alpha=(M*/SFR)/(t_H-1 Gyr), models predict a constantalpha~1 out to redshifts z=4+, while the observed M*-SFR relation indicatesthat alpha increases by X3 from z~2 until today. The low alpha at high-z notonly conflicts with models, but is also difficult to reconcile with otherobservations of high-z galaxies. Systematic biases could significantly affectmeasurements of M* and SFR, but detailed considerations suggest that none areobvious candidates to reconcile the discrepancy. A speculative solution isconsidered in which the stellar initial mass function (IMF) evolves towardsmore high-mass star formation at earlier epochs. Following Larson, a model isinvestigated in which the characteristic mass Mhat where the IMF turns overincreases with redshift. The observed and predicted M*-SFR evolution may bebrought into agreement if Mhat=0.5(1+z)^2 Mo out to z~2. Such evolution broadlymatches recent observations of cosmic stellar mass growth, and the resultingz=0 cumulative IMF is similar to the paunchy IMF favored by Fardal et al toreconcile the observed cosmic star formation history with present-day fossillight measures. [abridged]
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