If the Universe was dominated by a degenerate Fermi gas, would it have a maximum density in the distant past? This question is inspired by this one, but focusing on an aspect I found interesting and (in my opinion) was not fully addressed in the answers (which did, however, answer the question there).
The scenario I'm describing is probably not realistic cosmologically, but I think it should be answerable with standard GR/cosmology and quantum field theory -- or at least it should be possible to show this scenario is either impossible or outside the regime of validity of the effective field theory of gravity. Hopefully this doesn't cause any issues with this being about "mainstream physics," since I think it's a well-posed thought experiment that should be answerable with mainstream theoretical ideas.
Let's say the Universe consists of one matter field, which is a fermionic field. Suppose the fermions form a homogeneous and isotropic fluid with some energy density $\rho$ and pressure $p$, and therefore drive the expansion of the Universe via the Friedmann equations. If you like, assume these are relativistic fermions, but I am not too concerned about this -- feel free to make any reasonable assumption needed about the equation of state to answer the question.
Furthermore, let's suppose there is some initial but finite time $t_\star$ where the Fermi temperature is equal to or larger than the temperature of the fermion field, so the fermion fluid is very dense but cold, as in a neutron star. (Maybe my question has different answers for "equal to" and "larger than"?)
What would happen if we tried to evolve the Universe back to an earlier time before $t_\star$? On the one hand, based on the Friedman equations, I would expect the Universe to get denser until it hit the Big Bang singularity. On the other hand, by the Pauli exclusion principle, I would expect that the fermion gas would exert a degeneracy pressure that prevented them from reaching a smaller density.
 A: Evolving forward in time, energy input is required in order to compress a Fermi gas--this is because the energy is needed to bring the fermions to higher energy states so that they obey the Pauli Exclusion Principle. If the energy does not exist, then compression cannot be attained. This energy requirement manifests as a pressure, which prevents the compression of degenerate matter, unless you provide enough energy to squeeze it!
Looking backward in time, to reach an ever-smaller density, all that is required is an ever-larger initial energy. Evolving backward in time is essentially solving for the necessary initial conditions you must assume to posit such a Universe in the first place.
For relativistic particles, this is fine, since the Fermi temperature scales as $L^{-1}$ and the particle temperature will evolve similarly, since it will behave similarly to radiation (since it is relativistic).
Not too different from a radiation-dominated Universe, whose energy densities will also increase rapidly as you shrink the Universe.
Essentially, letting a spring go is different from compressing a spring. In this case, you are just letting a spring go and assuming it was already compressed by initial conditions.
No maximum density necessary.
