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.
