Cosmic happenstance: 24-$\mu$m selected, multi-component Herschel
  sources are line-of-sight projections
  In this paper, we investigate the physical associations between blended
far-infrared (FIR)-emitting galaxies, in order to identify the level of
line-of-sight projection contamination in the single-dish Herschel data.
Building on previous work, and as part of the Herschel Extragalactic Legacy
Project (HELP), we identify a sample of galaxies in the COSMOS field which are
found to be both FIR-bright (typically $\sim 15$ mJy) and blended within the
Herschel 250 $\mu$m beam. We identify a spectroscopic or photometric redshift
for each FIR-bright source. We conduct a joint probability distribution
analysis on the redshift probability density functions to determine the
fraction of the FIR sources with multiple FIR-bright counterparts which are
likely to be found at consistent ($\Delta z$ $< 0.01$) redshifts. We find that
only 3 (0.4 per cent) of the pair permutations between counterparts are $>50$
per cent likely to be at consistent redshifts. A majority of counterparts (72
per cent) have no overlap in their redshift probability distributions
whatsoever. This is in good agreement with the results of recent simulations,
which indicate that single-dish observations of the FIR sky should be strongly
contaminated by line of sight projection effects. We conclude that for our
sample of 3.6- and 24-$\mu$m selected, FIR-bright objects in the COSMOS field,
the overwhelming majority of multi-component FIR systems are line of sight
projections within the 18.1 arcsec Herschel beam, rather than physical
associations.
