HerMES: A Statistical Measurement of the Redshift Distribution of
  Herschel-SPIRE Sources Using the Cross-correlation Technique
  The wide-area imaging surveys with the {\it Herschel} Space Observatory at
sub-mm wavelengths have now resulted in catalogs of order one hundred thousand
dusty, star-burst galaxies. We make a statistical estimate of $N(z)$ using a
clustering analysis of sub-mm galaxies detected at each of 250, 350 and 500
$\mu$m from the Herschel Multi-tiered Extragalactic Survey (HerMES) centered on
the Bo\"{o}tes field. We cross-correlate {\it Herschel} galaxies against galaxy
samples at optical and near-IR wavelengths from the Sloan Digital Sky Survey
(SDSS), the NOAO Deep Wide Field Survey (NDWFS) and the Spitzer Deep Wide Field
Survey (SDWFS). We create optical and near-IR galaxy samples based on their
photometric or spectroscopic redshift distributions and test the accuracy of
those redshift distributions with similar galaxy samples defined with catalogs
of the Cosmological Evolution Survey (COSMOS), as the COSMOS field has superior
spectroscopy coverage. We model-fit the clustering auto and cross-correlations
of {\it Herschel} and optical/IR galaxy samples to estimate $N(z)$ and
clustering bias factors. The $S_{350} > 20$ mJy galaxies have a bias factor
varying with redshift as $b(z)=1.0^{+1.0}_{-0.5}(1+z)^{1.2^{+0.3}_{-0.7}}$.
This bias and the redshift dependence is broadly in agreement with galaxies
that occupy dark matter halos of mass in the range of 10$^{12}$ to 10$^{13}$
M$_{\sun}$. We find that the redshift distribution peaks around $z \sim 0.5$ to
1 for galaxies selected at 250 $\mu$m with an average redshift of $< z > = 1.8
\pm 0.2$. For 350 and 500 $\mu$m-selected SPIRE samples the peak shifts to
higher redshift, but the average redshift remains the same with a value of $1.9
\pm 0.2$.
