Implications of a spatially resolved main sequence for the size evolution of star-forming galaxies

Two currently debated problems in galaxy evolution, the fundamentally local or global nature of the main sequence of star formation and the evolution of the mass-size relation of star-forming galaxies (SFGs), are shown to be intimately related to each other. As a preliminary step, a growth function g is defined, which quantifies the differential change in half-mass radius per unit increase in stellar mass (g = d log R1/2/d log M) due to star formation. A general derivation shows that g = KΔ(sSFR)/sSFR, meaning that g is proportional to the relative difference in specific star formation rate between the outer and the inner half of a galaxy, with K a dimensionless structural factor for which handy expressions are provided. As an application, it is shown that galaxies obeying a fundamentally local main sequence also obey, to a good approximation, g γn, where γis the slope of the normalized local main sequence ($\mathrm{ sSFR} \,\, \propto \,\, \Sigma \star {-\gamma }$) and n is the Sersic index. An exact expression is also provided. Quantitatively, a fundamentally local main sequence is consistent with SFGs growing along a stationary mass-size relation, but inconsistent with the continuation at z = 0 of evolutionary laws derived at higher z. This demonstrates that either the main sequence is not fundamentally local, or the mass-size relation of SFGs has converged to an equilibrium state at some finite time in the past, or both.