| Summary: | Sub-milllimetre galaxies (SMGs) are some of the most luminous star-forming galaxies in the Universe, however their properties remain hard to determine due to the difficulty of identifying their optical\slash near-infrared counterparts. One of the key steps to determining the nature of SMGs is measuring a redshift distribution representative of the whole population. We do this by applying statistical techniques to a sample of 761 850$\mu$m sources from the SCUBA-2 Cosmology Legacy Survey observations of the UKIDSS Ultra-Deep Survey (UDS) Field. We detect excess galaxies around $> 98.4$ per cent of the 850$\mu$m positions in the deep UDS catalogue, giving us the first 850$\mu$m selected sample to have virtually complete optical\slash near-infrared redshift information. Under the reasonable assumption that the redshifts of the excess galaxies are representative of the SMGs themselves, we derive a median SMG redshift of $z = 2.05 \pm 0.03$, with 68 per cent of SMGs residing between $1.07 < z < 3.06$. We find an average of $1.52\pm 0.09$ excess $K$-band galaxies within 12 arc sec of an 850$\mu$m position, with an average stellar mass of $2.2\pm 0.1 \times 10^{10}$ M$_\odot$. While the vast majority of excess galaxies are star-forming, $8.0 \pm 2.1$ per cent have passive rest-frame colours, and are therefore unlikely to be detected at sub-millimetre wavelengths even in deep interferometry. We show that brighter SMGs lie at higher redshifts, and use our SMG redshift distribution -- along with the assumption of a universal far-infrared SED -- to estimate that SMGs contribute around 30 per cent of the cosmic star formation rate density between $0.5 < z < 5.0$.
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