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 ($g = d \log R_{1/2}/d \log M_\star$) due to star formation. A general derivation shows that $g = K \Delta(sSFR)/sSFR$, meaning that $g$ is proportional to the relative difference in specific star formation rate between the outer and 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, $g \simeq \gamma n$, where $\gamma$ is the slope of the normalized local main sequence ($sSFR \propto \Sigma_\star^{-\gamma}$) and $n$ 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 some finite time in the past, or both.