  The analysis of the core recoil momentum distribution of neutron-rich
isotopes of light exotic nuclei is performed within a model of the halo nuclei
described by a core and two neutrons dominated by the $s-$wave channel. We
adopt the renormalized three-body model with a zero-range force, that accounts
for the universal Efimov physics. This model is applicable to nuclei with large
two-neutron halos compared to the core size, and a neutron-core scattering
length larger than the interaction range. The halo wave function in momentum
space is obtained by using as inputs the two-neutron separation energy and the
energies of the singlet neutron-neutron and neutron-core virtual states. Within
our model, we obtain the momentum probability densities for the Borromean
exotic nuclei Lithium-11 ($^{11}$Li), Berylium-14 ($^{14}$Be) and Carbon-22
($^{22}$C). A fair reproduction of the experimental data was obtained in the
case of the core recoil momentum distribution of $^{11}$Li and $^{14}$Be,
without free parameters. By extending the model to $^{22}$C, the combined
analysis of the core momentum distribution and matter radius suggest (i) a
$^{21}$C virtual state well below 1 MeV; (ii) an overestimation of the
extracted matter $^{22}$C radius; and (iii) a two-neutron separation energy
between 100 and 400 keV.
