ALMA constraints on the faint millimetre source number counts and their contribution to the cosmic infrared background

We have analysed 18 ALMA continuum maps in Bands 6 and 7, with rms down to 7.8$\mu$Jy, to derive differential number counts down to 60$\mu$Jy and 100$\mu$Jy at $\lambda=$1.3 mm and $\lambda=$1.1 mm, respectively. The area covered by the combined fields is $\rm 9.5\times10^-4deg^2$ at 1.1mm and $\rm 6.6\times10^-4deg^2$ at 1.3mm. We improved the source extraction method by requiring that the dimension of the detected sources be consistent with the beam size. This method enabled us to remove spurious detections that have plagued the purity of the catalogues in previous studies. We detected 50 faint sources with S/N$>$3.5 down to 60$\mu$Jy, hence improving the statistics by a factor of four relative to previous studies. The inferred differential number counts are $\rm dN/d(Log_10S)=1\times10^5~deg^2$ at a 1.1 mm flux $S_\lambda = 1.1~mm = 130~\mu$Jy, and $\rm dN/d(Log_10S)=1.1\times10^5~deg^2$ at a 1.3 mm flux $\rm S_\lambda = 1.3~mm = 60~\mu$Jy. At the faintest flux limits, i.e. 30$\mu$Jy and 40$\mu$Jy, we obtain upper limits on the differential number counts of $\rm dN/d(Log_10S) < 7\times10^5~deg^2$ and $\rm dN/d(Log_10S)<3\times10^5~deg^2$, respectively. Our results provide a new lower limit to CIB intensity of 17.2$\rm Jy\ deg^-2$ at 1.1mm and of 12.9$\rm Jy\ deg^-2$ at 1.3mm. Moreover, the flattening of the integrated number counts at faint fluxes strongly suggests that we are probably close to the CIB intensity. Our data imply that galaxies with SFR$<40~M_\odot/yr$ certainly contribute less than 50% to the CIB while more than 50% of the CIB must be produced by galaxies with $\rm SFR>40~M_\odot/yr$. The differential number counts are in nice agreement with recent semi-analytical models of galaxy formation even as low as our faint fluxes. Consequently, this supports the galaxy evolutionary scenarios and assumptions made in these models.