The kiloparsec-scale star formation law at redshift 4: wide-spread, highly efficient star formation in the dust-obscured starburst galaxy GN20

We present high-resolution observations of the 880 $\mu$m (rest-frame FIR) continuum emission in the z$=$4.05 submillimeter galaxy GN20 from the IRAM Plateau de Bure Interferometer (PdBI). These data resolve the obscured star formation in this unlensed galaxy on scales of 0.3$^{\prime\prime}$$\times$0.2$^{\prime\prime}$ ($\sim$2.1$\times$1.3 kpc). The observations reveal a bright (16$\pm$1 mJy) dusty starburst centered on the cold molecular gas reservoir and showing a bar-like extension along the major axis. The striking anti-correlation with the HST/WFC3 imaging suggests that the copious dust surrounding the starburst heavily obscures the rest-frame UV/optical emission. A comparison with 1.2 mm PdBI continuum data reveals no evidence for variations in the dust properties across the source within the uncertainties, consistent with extended star formation, and the peak star formation rate surface density (119$\pm$8 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$) implies that the star formation in GN20 remains sub-Eddington on scales down to 3 kpc$^2$. We find that the star formation efficiency is highest in the central regions of GN20, leading to a resolved star formation law with a power law slope of $\Sigma_{\rm SFR}$ $\sim$ $\Sigma_{\rm H_2}^{\rm 2.1\pm1.0}$, and that GN20 lies above the sequence of normal star-forming disks, implying that the dispersion in the star formation law is not due solely to morphology or choice of conversion factor. These data extend previous evidence for a fixed star formation efficiency per free-fall time to include the star-forming medium on $\sim$kpc-scales in a galaxy 12 Gyr ago.