  We calculate neutron-neutron relative-energy distributions of $s$-wave
two-neutron ($2n$) halo nuclei using Halo Effective Field Theory (Halo EFT) at
leading order. At this order these systems are described by the $2n$ separation
energy, the neutron-core ($nc$) virtual-state energy and the neutron-neutron
($nn$) scattering length. We focus on knockout reactions where the removal of
the core is sudden, such that the final-state interactions are dominated by the
$nn$ interaction. We consider the neutron relative-energy distribution for the
nuclei $^{11}$Li, $^{14}$Be, $^{17}$B, $^{19}$B, and $^{22}$C. We show that the
ground-state neutron momentum distributions of all these nuclei stem from a
single curve, which can be obtained by taking both the neutron-core and
neutron-neutron interaction to the unitary limit. This universal description
can be extended to the final distribution measured in experiment by including
$nn$ final-state interactions via the approximate technique of enhancement
factors. For all the nuclei considered we find good agreement between the full
leading-order Halo EFT calculation and the universal prediction obtained in
this way. The universality of the ground-state momentum distribution in
two-neutron Borromean halos can thus be tested by dividing the experimental
results from sudden core knockout by the enhancement factor and comparing to
the unitary-limit prediction.
