Dusty, star forming galaxies contribute to a bright, currently unresolved cosmic far-infrared background. Deep Herschel-SPIRE images designed to detect and characterize the galaxies that comprise this background are highly confused, such that the bulk lies below the classical confusion limit. We analyze three fields from the HerMES programme in all three SPIRE bands (250, 350, and 500 microns); parameterized galaxy number count models are derived to a depth of ~2 mJy/beam, approximately 4 times the depth of previous analyses at these wavelengths, using a P(D) (probability of deflection) approach for comparison to theoretical number count models. Our fits account for 64, 60, and 43 per cent of the far-infrared background in the three bands. The number counts are consistent with those based on individually detected SPIRE sources, but generally inconsistent with most galaxy number counts models, which generically overpredict the number of bright galaxies and are not as steep as the P(D)-derived number counts. Clear evidence is found for a break in the slope of the differential number counts at low flux densities. Systematic effects in the P(D) analysis are explored. We find that the effects of clustering have a small impact on the data, and the largest identified systematic error arises from uncertainties in the SPIRE beam.
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