A complete distribution of redshifts for sub-millimetre galaxies in the
  SCUBA-2 Cosmology Legacy Survey UDS field
  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 $0.5 < z <
5.0$.
