How does the infinite density of the early universe or black holes not violate Fermi-Dirac statistics? When we talk about the Robertson-Walker scale factor in the early universe or the matter inside a black hole event horizon, there must be an arbitrarily large or infinite density. 
But how can this be possible if we consider Fermi-Dirac statistics? I mean suppose the scale factor decreases toward early times. Can we squeeze the matter as much as the degeneracy pressure allows us in order to put a large amount of matter inside a limited physical space? How can we imagine a typical galaxy inside a smaller space that is not expanded as much as today, let alone all the galaxies in the universe? The same is true about a black hole. How can we explain infinite density and growing matter inside a limited physical space? Even if we say that their momentum quantum states are respecting the statistics but again there are particles which occupy physical space, which is by assumption far smaller than the current epoch or outside of a black hole event horizon.
 A: At present the infinities of classical physics are "explained" by quantum mechanics.
All the classical 1/r^2 fields become infinite at r=0. Quantum mechanics with the energy levels not allowing to reach r=0  saves the day and we have atoms with electrons in quantized energy levels. Even free particles are saved by the Heisenberg uncertainty principle which introduces a basic fuzzines where the singularity existed.
The same logic is used in the Big Bang model, where quantized gravity is pulled in to model what was the original bang singularity.
In this answer you will see that quantum mechanics is used  to explain how galaxies evolve, in a nutshell from seeds of quantum mechanical fluctuations . 
As for statistics, in the beginning of the universe the inflatons are scalars, much later  the table of particles with their antiparticles are generated from the quantum mechanical soup.
Black holes have not been modeled to such an extent because we have no data for inside the black holes in the way we have observational data for the universe. Once gravity is definitively quantized ( at the moment it is all effective quantum field theory for gravity) I suppose people will tackle black holes too.
