An extended proto-cluster of galaxies surrounding SPT2349$-$56 at $z\,{=}\,4.3$

The $z\,{=}\,4.3$ overdensity SPT2349$-$56 has been proposed as one of the most actively star-forming proto-clusters known, and we have undertaken an extensive ALMA spectroscopic follow-up programme of this system to search for far infrared-luminous proto-cluster members via their [CII] 158$\,\mu$m and CO(4-3) line emission. In addition to robustly detecting the 14 previously published galaxies in this structure, we identify a further 15 associated galaxies at $z\,{=}\,4.3$. These proto-cluster members are distributed into a central core containing 23 galaxies extending out to 300$\,$kpc in diameter, and a northern extension, offset from the core by 400$\,$kpc and containing three galaxies. We discovered three additional galaxies in a red Herschel-SPIRE source 2$\,$Mpc from the main structure, suggesting the existence of many other sources at the same redshift as SPT2349$-$56 that are not yet detected in the limited coverage of our data. An analysis of the velocity distribution within the core indicates that this region may be a nearly virialized structure with a mass of (9$\pm$5)$\,{\times}\,$10$^{12}\,$M$_{\odot}$, while the two offset substructures show significant velocity offsets from the central structure. We calculate the [CII] and far-infrared number counts, and find evidence for a break in the [CII] luminosity function around 4$\,{\times}\,10^9\,$L$_{\odot}$. We estimate the average SFR density within the region of SPT2349$-$56 containing single-dish emission (a proper diametre of 720$\,$kpc), assuming spherical symmetry, to be roughly 40,000$\,$M$_{\odot}\,$yr$^{-1}\,$Mpc$^{-3}$; this is an order of magnitude greater than the most extreme examples seen in simulations.