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 $\rm 9.5\times10^{-4}deg^2$ at 1.1mm and $\rm
6.6\times10^{-4}deg^{2}$ 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 $\rm dN/d(Log_{10}S)=1\times10^5~deg^2$ at a 1.1 mm flux $S_{\lambda
= 1.1~mm} = 130~\mu$Jy, and $\rm dN/d(Log_{10}S)=1.1\times10^5~deg^2$ 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 $\rm dN/d(Log_{10}S) < 7\times10^5~deg^2$ and $\rm
dN/d(Log_{10}S)<3\times10^5~deg^2$, 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.
