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The star-forming main sequence and the contribution of dust-obscured star formation since $z\sim4$ from the FUV+IR luminosity functions

An analytical approach is proposed to study the evolution of the star-forming galaxy (SFG) main sequence (MS) and the fraction of dust-obscured SF up to $z\sim4$. Far-ultraviolet (FUV) and infrared (IR) star formation rates, SFRs, are described as conditional probability functions of $M_{\ast}$. We convolve them with the galaxy stellar mass function (GSMF) of SFGs to derive the FUV and IR LFs. The 2 SF modes formalism is used to describe starburst galaxies. By fitting observed FUV and IR LFs, the parametrization of SFR$_{\rm FUV}-M_{\ast}$ and SFR$_{\rm IR}-M_{\ast}$ are constrained. Our derived SFR$_{\rm FUV+IR}-M_{\ast}$ reproduces the evolution of the MS as compared to other observational inferences. At any redshift, we find that the sSFR$_{\rm FUV+IR}-M_{\ast}$ relation for MS SFGs approaches to a power law at the high-mass end. At lower masses, it bends and eventually the slope sign changes from negative to positive at very low masses. At $z\sim0$, this change of sign is at $M_{\ast}\sim5\times10^{8}{\rm M}_{\odot}$ close to dust-obscured SF regime, $M_{\ast}\sim6\times10^{8}{\rm M}_{\odot}$. The slope sign change is related to the knee of the FUV LF. Our derived dust-obscured fractions agree with previous determinations at $0\leq z\leq2.5$. Dust-obscured fractions depend strongly on mass with almost no dependence with redshift at $z\gtrsim1.2$. At $z\lesssim0.75$ high-mass galaxies become more "transparent" compared to their high redshift counterparts. On the opposite, low- and intermediate-mass galaxies have become more obscured by dust. The joint evolution of the GSMF and the FUV and IR LFs is a promising approach to study mass growth and dust formation/destruction mechanisms.