The SCUBA-2 Cosmology Legacy Survey: The EGS deep field I - Deep number counts and the redshift distribution of the recovered Cosmic Infrared Background at 450 and 850 um

We present deep observations at 450 um and 850 um in the Extended Groth Strip field taken with the SCUBA-2 camera mounted on the James Clerk Maxwell Telescope as part of the deep SCUBA-2 Cosmology Legacy Survey (S2CLS), achieving a central instrumental depth of $\sigma_{450}=1.2$ mJy/beam and $\sigma_{850}=0.2$ mJy/beam. We detect 57 sources at 450 um and 90 at 850 um with S/N > 3.5 over ~70 sq. arcmin. From these detections we derive the number counts at flux densities $S_{450}>4.0$ mJy and $S_{850}>0.9$ mJy, which represent the deepest number counts at these wavelengths derived using directly extracted sources from only blank-field observations with a single-dish telescope. Our measurements smoothly connect the gap between previous shallower blank-field single-dish observations and deep interferometric ALMA results. We estimate the contribution of our SCUBA-2 detected galaxies to the cosmic infrared background (CIB), as well as the contribution of 24 um-selected galaxies through a stacking technique, which add a total of $0.26\pm0.03$ and $0.07\pm0.01$ MJy/sr, at 450 um and 850 um, respectively. These surface brightnesses correspond to $60\pm20$ and $50\pm20$ per cent of the total CIB measurements, where the errors are dominated by those of the total CIB. Using the photometric redshifts of the 24 um-selected sample and the redshift distributions of the submillimetre galaxies, we find that the redshift distribution of the recovered CIB is different at each wavelength, with a peak at $z\sim1$ for 450 um and at $z\sim2$ for 850um, consistent with previous observations and theoretical models.