Insights on star-formation histories and physical properties of 1.2 ≤ z ≲ 4 Herschel-detected galaxies

Aims. We aim to test the impact of using variable star-forming histories (SFHs) and the IR luminosity as a constrain on the physical parameters of high redshift dusty star-forming galaxies. We explore in particular the properties (SFHs, ages, timescales) of galaxies depending on their belonging to the "main sequence" of star-forming galaxies (MS). Methods. We performed spectral energy distribution (SED) fitting of the UV-to-NIR and FIR emissions of a moderately large sample of GOODS-Herschel galaxies, for which rich multi-wavelength, optical to IR observations are available. We tested different SFHs and the impact of imposing energy conservation in the SED fitting process, to help with issues like the age-extinction degeneracy and produce SEDs consistent with observations. Results. Our simple models produce well constrained SEDs for the broad majority of the sample (84%), with the notable exception of the very high L-IR end, for which we have indications that the energy conservation hypothesis cannot hold true for a single component population approach. We observe trends in the preferences in SFHs among our sources depending on stellar mass M and z. Trends also emerge in the characteristic timescales of the SED models depending on the location on the SFR - M diagram. We show that whilst using the same available observational data, we can produce galaxies less star-forming than classically inferred, if we allow rapidly declining SFHs, while properly reproducing their observables. These sources, representing 7% of the sample, can be post-starbursts undergoing quenching, and their SFRs are potentially overestimated if inferred from their LIR. Based on the trends observed in the rising SFH fits we explore a simple evolution model for stellar mass build-up over the considered time period. Conclusions. Our approach successfully breaks the age-extinction degeneracy, and enables to evaluate properly the SFRs of the sources in the SED fitting process. Fitting without the IR constrain leads to a strong preference for declining SFHs, while its inclusion increases the preference of rising SFHs, more so at high z, in tentative agreement with the cosmic star-formation history (CSFH), although this result suffers from poor statistics. Keeping in mind that the sample is biased toward high luminosities and intense star formation, the evolution shaped by our model appears as both bursty (in its early stages) and steady-lasting (later on). The SFH of the sample considered as a whole follows the CSFH with a surprisingly small scatter, and is compatible with other studies supporting that the more massive galaxies have built most of their mass earlier than lower mass galaxies.