the radiation field $\langle U \rangle$ to vary by $\pm$0.6dex (3$\sigma$ of the scatter used in the @Bethermin2012c model). These criteria of two detections at different wavelengths and the high reliability of the detections prevent biasing of the starbursts towards positive fluctuations of the noise in the maps and limit the flux boosting effect. We then estimated the SFR from the infrared luminosity, L$_{\rm IR}$, using the @Kennicutt1998 relation. We performed a first analysis using the same evolution of the main-sequence (sSFR$_{\rm MS}$ versus z) as in @Bethermin2012c to select sSFR$>$10 sSFR$_{\rm MS}$ objects. We then fit the measured evolution of the main-sequence found by a first stacking analysis (see Sect.\[sect:stacking\] and Sect.\[sect:sedfit\]) to prepare the final sample for our analysis. We could have chosen a lower sSFR cut corresponding to 4 times the value at the center of the main-sequence as in @Rodighiero2011, but the sample would be incomplete at z$>$1 because of the flux limit of the infrared catalogs.\

![\[SBcomp\] The thick red solid line represents the luminosity limit corresponding to a criterion of a 5$\sigma$ detection in at least two *Herschel* bands. The other solid lines are the limits for a detection at only one given wavelength 