  We derive the SFH of MS galaxies showing how the SFH peak of a galaxy depends
on its seed mass at e.g. z=5. Following the MS, galaxies undergo a drastic slow
down of their stellar mass growth after reaching the peak of their SFH.
According to abundance matching, these masses correspond to hot and massive DM
halos which state could results in less efficient gas inflows on the galaxies
and thus could be at the origin of the limited stellar mass growth. As a
result, galaxies on the MS can enter the passive region of the UVJ diagram
while still forming stars. The ability of the classical analytical SFHs to
retrieve the SFR of galaxies from SED fitting is studied. Due to mathematical
limitations, the exp-declining and delayed SFH struggle to model high SFR which
starts to be problematic at z>2. The exp-rising and log-normal SFHs exhibit the
opposite behavior with the ability to reach very high SFR, and thus model
starburst galaxies, but not low values such as those expected at low redshift
for massive galaxies. We show that these four analytical forms recover the SFR
of MS galaxies with an error dependent on the model and the redshift. They are,
however, sensitive enough to probe small variations of SFR within the MS but
all the four fail to recover the SFR of rapidly quenched galaxies. However,
these SFHs lead to an artificial gradient of age, parallel to the MS which is
not exhibited by a simulated sample. This gradient is also produced on real
data, using a sample of GOODS-South galaxies at 1.5<z<1.2. We propose a SFH
composed of a delayed form to model the bulk of stellar population plus a
flexibility in the recent SFH. This SFH provides very good estimates of the SFR
of MS, starbursts, and rapidly quenched galaxies at all z. Furthermore, used on
the GOODS-South sample, the age gradient disappears, showing its dependency on
the SFH assumption made to perform the SED fitting.
