  [abridged] Understanding how the infalling gas redistribute most of its
initial angular momentum inherited from prestellar cores before reaching the
stellar embryo is a key question. Disk formation has been naturally considered
as a possible solution to this "angular momentum problem". However, how the
initial angular momentum of protostellar cores is distributed and evolves
during the main accretion phase and the beginning of disk formation has largely
remained unconstrained up to now. In the framework of the IRAM CALYPSO survey,
we used high dynamic range C$^{18}$O (2-1) and N$_2$H$^+$ (1-0) observations to
quantify the distribution of specific angular momentum along the equatorial
axis in a sample of 12 Class 0 protostellar envelopes from scales ~50 to 10000
au. The radial distributions of specific angular momentum in the CALYPSO sample
suggest two distinct regimes within protostellar envelopes: the specific
angular momentum decreases as $j \propto r^{1.6 \pm 0.2}$ down to ~1600 au and
then tends to become relatively constant around 6 $\times$ 10$^{-4}$ km
s$^{-1}$ pc down to ~50 au. The values of specific angular momentum measured in
the inner Class 0 envelopes, namely that of the material directly involved in
the star formation process ($<$1600 au), is on the same order of magnitude as
what is inferred in small T-Tauri disks. Thus, disk formation appears to be a
direct consequence of angular momentum conservation during the collapse. Our
analysis reveals a dispersion of the directions of velocity gradients at
envelope scales $>$1600 au, suggesting that they may not be related to
rotational motions of the envelopes. We conclude that the specific angular
momentum observed at these scales could find its origin in core-forming motions
(infall, turbulence) or trace an imprint of the initial conditions for the
formation of protostellar cores.
