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.