2
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

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.

Improve this answer
$\endgroup$
5
  • $\begingroup$ Thanks for the response! I definitely agree I am postulating a special initial condition, and the answer might well be that whatever is needed to get that initial condition (which I left vague) is unphysical. But I'm still left with a question, which is -- as you said, evolving backward is like solving for what initial condition I needed to get to where I am now. So, what came before my postulated state at $t_\star$ where the Universe's temperature is below the Fermi temperature? $\endgroup$
    – Andrew
    Commented Sep 1, 2021 at 3:34
  • $\begingroup$ In your spring example, what happened "before" max compression is that the spring was compressing -- by analogy, does this mean the Universe must have been contracting before it hit this degenerate Fermi state? (I doubt the spring example explains what's going on, but I'm using it to hopefully clarify what I'm asking). $\endgroup$
    – Andrew
    Commented Sep 1, 2021 at 3:35
  • 1
    $\begingroup$ What came before your postulated state was simply a Universe with higher energy density, as in all cosmological models wherein the Universe starts at higher energy density which decreases as it expands. What happened before was the spring started in a compressed state. Watching the spring forward in time, it simply expands. $\endgroup$
    – Alwin
    Commented Sep 1, 2021 at 3:38
  • 1
    $\begingroup$ I think I see what you are saying. Can I rephrase your argument like this? "If, at some time $t_\star$, the fermion-filled Universe had a temperature equal to the Fermi temperature, then we can use the Friedman equations as usual to evolve the scale factor backward in time. Since both the temperature and the Fermi temperature will have the same scaling with time, the ratio of the Fermi temperature and actual temperature will never change. It's impossible to have the temperature less than the Fermi temperature, so this situation never arises." I think that makes sense to me! $\endgroup$
    – Andrew
    Commented Sep 1, 2021 at 12:34
  • 1
    $\begingroup$ Yes, and moreover, even if the matter were degenerate, it would be okay to evolve backwards in time, since you can just posit that the material has a high initial energy density so that it can reach the desired density. Which is no different than the high energy density requirement for a Universe full of only photons. So it will run into the same mysteries at the Planck scale, but that aside, no real problem. $\endgroup$
    – Alwin
    Commented Sep 1, 2021 at 18:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.