Multi-wavelength de-blended Herschel view of the statistical properties of dusty star-forming galaxies across cosmic time

Aims. We study the statistical properties of dusty star-forming galaxies across cosmic time, such as their number counts, luminosity functions (LF), and the dust-obscured star formation rate density (SFRD). Methods. We used the most recent de-blended Herschel catalogue in the COSMOS field to measure the number counts and LFs at far-infrared (FIR) and sub-millimetre (sub-mm) wavelengths. The de-blended catalogue was generated by combining the Bayesian source extraction tool XID+ and an informative prior derived from the associated deep multi-wavelength photometric data. Results. Through our de-confusion technique and based on the deep multi-wavelength photometric information, we are able to achieve more accurate measurements while at the same time probing roughly ten times below the Herschel confusion limit. Our number counts at 250 mu m agree well with previous Herschel studies. However, our counts at 350 and 500 mu m are below previous Herschel results because previous Herschel studies suffered from source confusion and blending issues. Our number counts at 450 and 870 mu m show excellent agreement with previous determinations derived from single-dish and interferometric observations. Our measurements of the LF at 250 mu m and the total IR LF agree well with previous results in the overlapping redshift and luminosity range. The increased dynamic range of our measurements allows us to better measure the faint-end of the LF and measure the dust-obscured SFRD out to z similar to 6. We find that the fraction of obscured star formation activity is at its highest (>80%) around z similar to 1. We do not find a shift of balance between z similar to 3 and z similar to 4 in the SFRD from being dominated by unobscured star formation at higher redshift to obscured star formation at lower redshift. However, we do find 3 < z < 4 to be an interesting transition period as the portion of the total SFRD that is obscured by dust is significantly lower at higher redshifts.