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.
