ALMA maps the Star-Forming Regions in a Dense Gas Disk at z~3

We exploit long-baseline ALMA sub-mm observations of the lensed star-forming galaxy SDP 81 at z=3.042 to investigate the properties of inter-stellar medium on scales of 50-100pc. The kinematics of the CO gas within this system are well described by a rotationally-supported disk with an inclination-corrected rotation speed, v=320+/-20km/s and a dynamical mass of M=(3.5+/-1.0)x10^10Mo within a radius of 1.5 kpc. The disk is gas rich and unstable, with a Toomre parameter, Q=0.30+/-0.10 and so should collapse in to star-forming regions with Jeans length L_J~130pc. We identify five star-forming regions within the ISM on these scales and show that their scaling relations between luminosity, line-widths and sizes are significantly offset from those typical of molecular clouds in local Galaxies (Larson's relations). These offsets are likely to be caused by the high external hydrostatic pressure for the interstellar medium (ISM), P/kB=(40+/-20)x10^7K/cm3, which is ~10,000x higher than the typical ISM pressure in the Milky Way. The physical conditions of the star-forming ISM and giant molecular clouds appears to be similar to the those found in the densest environments in the local Universe, such as those in the Galactic center.