Probing star formation in the dense environments of z~1 lensing halos
  aligned with dusty star-forming galaxies detected with the South Pole
  Telescope
  We probe star formation in the environments of massive
$\sim10^{13}\,M_{\odot}$ dark matter halos at redshifts of $z$$\sim$$1$. This
star formation is linked to a sub-millimetre clustering signal which we detect
in maps of the Planck High Frequency Instrument that are stacked at the
positions of a sample of high-redshift ($z$$>$$2$) strongly-lensed dusty
star-forming galaxies (DSFGs) selected from the South Pole Telescope (SPT) 2500
deg$^2$ survey. The clustering signal has sub-millimetre colours which are
consistent with the mean redshift of the foreground lensing halos
($z$$\sim$$1$). We report a mean excess of star formation rate (SFR) compared
to the field, of $(2700\pm700)\,M_{\odot}\,{yr}^{-1}$ from all galaxies
contributing to this clustering signal within a radius of 3.5' from the SPT
DSFGs. The magnitude of the Planck excess is in broad agreement with
predictions of a current model of the cosmic infrared background. The model
predicts that 80$\%$ of the excess emission measured by Planck originates from
galaxies lying in the neighbouring halos of the lensing halo. Using Herschel
maps of the same fields, we find a clear excess, relative to the field, of
individual sources which contribute to the Planck excess. The mean excess SFR
compared to the field is measured to be ($370\pm40)$$\,M_{\odot}\,{yr}^{-1}$
per resolved, clustered source. Our findings suggest that the environments
around these massive $z$$\sim$$1$ lensing halos host intense star formation out
to about $2\,$Mpc. The flux enhancement due to clustering should also be
considered when measuring flux densities of galaxies in Planck data.
