  We offer a new, physically transparent argument for the existence of the
critical, universal maximum matter density in loop quantum cosmology for the
case of a flat Friedmann-Lemaitre-Robertson-Walker cosmology with scalar
matter. The argument is based on the existence of a sharp exponential
ultraviolet cutoff in momentum space on the eigenfunctions of the quantum
cosmological dynamical evolution operator (the gravitational part of the
Hamiltonian constraint), attributable to the fundamental discreteness of
spatial volume in loop quantum cosmology. The existence of the cutoff is proved
directly from recently found exact solutions for the eigenfunctions for this
model. As a consequence, the operators corresponding to the momentum of the
scalar field and the spatial volume approximately commute. The ultraviolet
cutoff then implies that the scalar momentum, though not a bounded operator, is
in effect bounded on subspaces of constant volume, leading to the upper bound
on the expectation value of the matter density. The maximum matter density is
universal (i.e. independent of the quantum state) because of the linear scaling
of the cutoff with volume. These heuristic arguments are supplemented by a new
proof in the volume representation of the existence of the maximum matter
density. The techniques employed to demonstrate the existence of the cutoff
also allow us to extract the large volume limit of the exact eigenfunctions,
confirming earlier numerical and analytical work showing that the
eigenfunctions approach superpositions of the eigenfunctions of the
Wheeler-DeWitt quantization of the same model. We argue that generic (not just
semiclassical) quantum states approach symmetric superpositions of expanding
and contracting universes.
