  We explore star-formation histories (SFHs) of galaxies based on the evolution
of the star-formation rate stellar mass relation (SFR-M*). Using data from the
FourStar Galaxy Evolution Survey (ZFOURGE) in combination with far-IR imaging
from the Spitzer and Herschel observatories we measure the SFR-M* relation at
0.5 < z < 4. Similar to recent works we find that the average infrared SEDs of
galaxies are roughly consistent with a single infrared template across a broad
range of redshifts and stellar masses, with evidence for only weak deviations.
We find that the SFR-M* relation is not consistent with a single power-law of
the form SFR ~ M*^a at any redshift; it has a power-law slope of a~1 at low
masses, and becomes shallower above a turnover mass (M_0) that ranges from
10^9.5 - 10^10.8 Msol, with evidence that M_0 increases with redshift. We
compare our measurements to results from state-of-the-art cosmological
simulations, and find general agreement in the slope of the SFR-M* relation
albeit with systematic offsets. We use the evolving SFR-M* sequence to generate
SFHs, finding that typical SFRs of individual galaxies rise at early times and
decline after reaching a peak. This peak occurs earlier for more massive
galaxies. We integrate these SFHs to generate mass-growth histories and compare
to the implied mass-growth from the evolution of the stellar mass function. We
find that these two estimates are in broad qualitative agreement, but that
there is room for improvement at a more detailed level. At early times the SFHs
suggest mass-growth rates that are as much as 10x higher than inferred from the
stellar mass function. However, at later times the SFHs under-predict the
inferred evolution, as is expected in the case of additional growth due to
mergers.
