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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.

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marked as duplicate by Ben Crowell, Kyle Kanos, John Rennie black-holes Sep 4 '18 at 10:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Infinities are invariably a sign of absurdity in the theory or in how it is being applied to a particular situation. Therefore, we explain infinite density as meaning that the application of the theory is incorrect, and that either a black hole has no density and must be reasoned about in some other way, or it has a density that is not properly described by the theory which predicts the infinity in that case. $\endgroup$ – Steve Feb 2 '18 at 11:27
  • $\begingroup$ What about the early universe conditions? In standard model of cosmology it is said that the expanding universe model based on scale factor makes sense down to a few moments of the big bang. How we imagine a whole galaxy of matter inside a tiny space? $\endgroup$ – user56963 Feb 2 '18 at 11:48
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    $\begingroup$ Might be well "the reason" for overwhelming expansion. Not sure about scales involved but the fermions were to emerge. In BBs I have no idea. $\endgroup$ – Alchimista Feb 2 '18 at 11:54
  • $\begingroup$ There is little on the frontiers of science that "makes sense" - there are two types of models in the scientific world, those that are considered complete, internally consistent, and are in everyday use, and which are already falsified, and those that don't yet make full sense but are being worked on (toward making them complete, at which point they can be decisively falsified)! $\endgroup$ – Steve Feb 2 '18 at 12:36
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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.

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  • $\begingroup$ They do have a surface area proportional to their entropy though, and can be thought about holographically as a holographic surface sciencealert.com/… $\endgroup$ – CriglCragl Feb 2 '18 at 14:36
  • $\begingroup$ What is a the quantum mechanical soup? $\endgroup$ – user56963 Feb 2 '18 at 14:58
  • $\begingroup$ In the diagram in this link en.wikipedia.org/wiki/Inflation_(cosmology)( which I have used in my other answer to which i linked in this one) it is before 10^-32 seconds where it is the scalar inflatons that dominate, due to the large energy density. The slowly the elementary particles of the standard model are created and destroyed in pair productions until 1ms when the universe has expanded enough for the protons to form. etc $\endgroup$ – anna v Feb 2 '18 at 15:37
  • $\begingroup$ @CriglCragl sure. I am talking of a model analogous to the Big Bang model for what is happening at the singularity. $\endgroup$ – anna v Feb 2 '18 at 15:40
  • $\begingroup$ One can also point out that as a gas expands adiabatically, its degree of degeneracy does not change. The Big Bang expansion is much like an adiabatic expansion, as there is no heat transfer. Running time backward, this also means that degeneracy does not increase no matter how far back you go and how high the densities become (until the physics undergoes some fundamental change as mentioned above). It turns out that the way to increase the degree of degeneracy of a gas is to remove heat from it, not to compress it adiabatically like a backward-running Big Bang. $\endgroup$ – Ken G Mar 15 '18 at 14:38