ABRIGED Herschel/SPIRE has provided confusion limited maps of deep fields at 250, 350, and 500um, as part of the HerMES survey. Due to confusion, only a small fraction of the Cosmic Infrared Background can be resolved into individually-detected sources. Our goal is to produce deep galaxy number counts and redshift distributions below the confusion limit, which we then use to place strong constraints on the origins of the cosmic infrared background and on models of galaxy evolution. We individually extracted the bright SPIRE with a method using the positions, the flux densities, and the redshifts of the 24um sources as a prior, and derived the number counts and redshift distributions of the bright SPIRE sources. For fainter SPIRE sources, we reconstructed the number counts and the redshift distribution below the confusion limit using the deep 24um catalogs associated with photometric redshift and information provided by the stacking of these sources into the deep SPIRE maps. Finally, by integrating all these counts, we studied the contribution of the galaxies to the CIB as a function of their flux density and redshift. Through stacking, we managed to reconstruct the source counts per redshift slice down to ~2 mJy in the three SPIRE bands, which lies about a factor 10 below the 5sigma confusion limit. None of the pre-existing population models are able to reproduce our results at better than 3sigma. Finally, we extrapolate our counts to zero flux density in order to derive an estimate of the total contribution of galaxies to the CIB, finding 10.1, 6.5, and 2.8 nW/m2/sr at 250, 350, and 500um, respectively. These values agree well with FIRAS absolute measurements, suggesting our number counts and their extrapolation are sufficient to explain the CIB. Finally, combining our results with other works, we estimate the energy budget contained in the CIB between 8 and 1000um: 26 nW/m2/sr.
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