Understanding the formation mechanisms of protoplanetary disks and multiple systems, and their pristine properties, is a key question for modern astrophysics. The properties of the youngest disks, embedded in rotating infalling protostellar envelopes, have largely remained unconstrained up to now. In the framework of the IRAM-PdBI CALYPSO survey, we have obtained sub-arcsecond observations of the dust continuum emission at 231 GHz and 94 GHz, for a sample of 16 solar-type Class 0 protostars. In an attempt to identify disk-like structures embedded at small scales in the protostellar envelopes, we model the dust continuum emission visibility profiles using both Plummer-like envelope models and envelope models including additional Gaussian disk-like components. Our analysis shows that in the CALYPSO sample, 11 of the 16 Class 0 protostars are better reproduced by models including a disk-like dust continuum component contributing to the flux at small scales, but less than 25% of these candidate protostellar disks are resolved at radii > 60 au. Including all available literature constraints on Class 0 disks at subarcsecond scales, we show that our results are representative: most (> 72% in a sample of 26 protostars) Class 0 protostellar disks are small and emerge only at radii < 60 au. Our multiplicity fraction at scales 100-5000 au is in global agreement with the multiplicity properties of Class I protostars at similar scales. We confront our observational constraints on the disk size distribution in Class 0 protostars to the typical disk properties from protostellar formation models. Because they reduce the centrifugal radius, and produce a disk size distribution peaking at radii <100 au during the main accretion phase, magnetized models of rotating protostellar collapse are favored by our observations.
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