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.
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