The Herschel Multi-Tiered Extragalactic Survey: SPIRE-mm Photometric
  Redshifts
  We investigate the potential of submm-mm and submm-mm-radio photometric
redshifts using a sample of mm-selected sources as seen at 250, 350 and 500
{\mu}m by the SPIRE instrument on Herschel. From a sample of 63 previously
identified mm-sources with reliable radio identifications in the GOODS-N and
Lockman Hole North fields 46 (73 per cent) are found to have detections in at
least one SPIRE band. We explore the observed submm/mm colour evolution with
redshift, finding that the colours of mm-sources are adequately described by a
modified blackbody with constant optical depth {\tau} = ({\nu}/{\nu}0)^{\beta}
where {\beta} = +1.8 and {\nu}0 = c/100 {\mu}m. We find a tight correlation
between dust temperature and IR luminosity. Using a single model of the dust
temperature and IR luminosity relation we derive photometric redshift estimates
for the 46 SPIRE detected mm-sources. Testing against the 22 sources with known
spectroscopic, or good quality optical/near-IR photometric, redshifts we find
submm/mm photometric redshifts offer a redshift accuracy of |z|/(1+z) = 0.16 (<
|z| >= 0.51). Including constraints from the radio-far IR correlation the
accuracy is improved to |z|/(1 + z) = 0.15 (< |z| >= 0.45). We estimate the
redshift distribution of mm-selected sources finding a significant excess at z
> 3 when compared to ~ 850 {\mu}m selected samples.
