Planck intermediate results. LIII. Detection of velocity dispersion from
  the kinetic Sunyaev-Zeldovich effect
  Using the ${\it Planck}$ full-mission data, we present a detection of the
temperature (and therefore velocity) dispersion due to the kinetic
Sunyaev-Zeldovich (kSZ) effect from clusters of galaxies. To suppress the
primary CMB and instrumental noise we derive a matched filter and then convolve
it with the ${\it Planck}$ foreground-cleaned `${\tt 2D-ILC\,}$' maps. By using
the Meta Catalogue of X-ray detected Clusters of galaxies (MCXC), we determine
the normalized ${\it rms}$ dispersion of the temperature fluctuations at the
positions of clusters, finding that this shows excess variance compared with
the noise expectation. We then build an unbiased statistical estimator of the
signal, determining that the normalized mean temperature dispersion of $1526$
clusters is $\langle \left(\Delta T/T \right)^{2} \rangle = (1.64 \pm 0.48)
\times 10^{-11}$. However, comparison with analytic calculations and
simulations suggest that around $0.7\,\sigma$ of this result is due to cluster
lensing rather than the kSZ effect. By correcting this, the temperature
dispersion is measured to be $\langle \left(\Delta T/T \right)^{2} \rangle =
(1.35 \pm 0.48) \times 10^{-11}$, which gives a detection at the $2.8\,\sigma$
level. We further convert uniform-weight temperature dispersion into a
measurement of the line-of-sight velocity dispersion, by using estimates of the
optical depth of each cluster (which introduces additional uncertainty into the
estimate). We find that the velocity dispersion is $\langle v^{2} \rangle
=(123\,000 \pm 71\,000)\,({\rm km}\,{\rm s}^{-1})^{2}$, which is consistent
with findings from other large-scale structure studies, and provides direct
evidence of statistical homogeneity on scales of $600\,h^{-1}{\rm Mpc}$. Our
study shows the promise of using cross-correlations of the kSZ effect with
large-scale structure in order to constrain the growth of structure.
