  We analyze high-resolution (400pc) 220GHz continuum and CO(2-1) ALMA
observations of a representative sample of 22 local (z<0.165) ULIRG systems (32
individual nuclei) as part of the "Physics of ULIRGs with MUSE and ALMA" (PUMA)
project. The deconvolved half-light radii of the 220GHz continuum sources are
between <60-350 pc (median 90pc). We associate these regions with the regions
emitting the bulk of the infrared luminosity. The good agreement, within a
factor of 2, between the 220GHz fluxes and the extrapolation of the infrared
gray-body, and the small synchrotron and free-free contributions support this
assumption. The cold molecular gas emission sizes are 60-700 pc and are on
average 2.6 times larger than the continuum. We derive L_IR and cold molecular
gas surface densities: log Sigma(L_IR)=11.5-14.3 Lsun/kpc^2 and log
Sigma(H2)=2.9-4.2 Msun/pc^2. Assuming that the L_IR is produced by
star-formation, this corresponds to median Sigma(SFR)=2500 Msun/yr/kpc^2 which
would imply extremely short depletion times, <1-15 Myr, and unphysical SF
efficiencies >1 for 70% of the sample. Therefore, this favors the presence of
obscured AGN that could dominate the L_IR. We also classify the ULIRG nuclei in
two groups: (a) compact nuclei (r<130 pc) with high mid-IR excess emission
found in optically classified AGN; and (b) nuclei following a relation with
decreasing mid-IR excess for decreasing r. The majority, 65%, of the
interacting nuclei lie in the low end (<130 pc) of this relation, while only
25% of the advanced mergers do so. This suggests that in the early stages of
the interaction, the activity occurs in more compact and dust-obscured regions
than in advanced merger stages. Approximately two thirds of the nuclei are
above the Eddington limit which is consistent with the detection of massive
outflows in local ULIRGs and the potential role of radiation pressure in the
launching process.
