# A universe with $\rho_{\text{vac}}>0$ but $p_{\text{vac}}=0$: how would it be like? Would it be stable?

The vacuum energy and vacuum pressure is $$\rho_{\text{vac}} = -p_{\text{vac}}$$ due to the cosmological constant in our universe.

I try to think about another universe, where $$\rho_{\text{vac}}$$ does not equal $$-p_{\text{vac}}$$. How would this universe be like?

It would be stable due to $$p_{\text{vac}}$$ set to zero - apart from gravitational forces, isn't it?

Or would the gravitation of the vacuum through $$\rho_{\text{vac}}$$ lead to a compression of the whole thing? But T00 is everywhere the same, so it should be like a constant field which leads to no forces, or?

• Hello! I have edited your question using MathJax (LaTeX) math typesetting. For future questions, you can refer to MathJax basic tutorial and quick reference. Thanks! Nov 20, 2021 at 9:49
• Thank you! I will have a look at this! Nov 20, 2021 at 10:01
• When you set $\rho_{\Lambda} \equiv \Omega_{\Lambda}=0$, you end up with a matter dominated universe. So the universe would be stable. Try to look at the solution for the matter dominated universe in any cosmological book. Nov 21, 2021 at 16:31
• I don' t understand this, SeVenVo1d. Do you mean if I set p=0 instead of rho=0? Nov 22, 2021 at 6:18

There are many possible matter models in General Relativity. In Cosmology, a family of simple models are obtained by using perfect fluids and the assumption that $$p = w \rho$$ for some $$w$$ that in simple models is taken to be a constant. In this framework, notice your model consists of taking $$w = 0$$.

This is actually a fairly common matter model, which we often refer to as "dust" or "cold matter". It is very well studied and you can find details about it on most textbooks on General Relativity and/or Cosmology. See, e.g., Sec. 8.3 onward of Sean Carroll's, Sec. 5.3 of Barbara Ryden's, Chap. 5 of Wald's, and so on. Notice that the accelerated expansion of the Universe was discovered in the late 90's, so older books (such as Wald's) are a bit outdated.

Changing the matter models leads to many changes in the geometry. For example, matter-filled Universes might recollapse into a Big Crunch. This might depend on the spatial curvature of the Universe, as discussed on the references I mentioned, but I recall that this does hold if the Universe is spatially a $$3$$-sphere, which implies the Universe being finite.

In other words, yes, at least in some cases the positive energy of everything leads the Universe to a recollapse, pulling everything together back into a singularity. In some cases (I'm not sure whether in all of them) this happens just like a reverse Big Bang: the Universe reaches a maximum size and then, in large scale, things behave just as if you were rewinding the tape, as you see the Universe shrink again and collapse onto itself.

Notice, however, that $$w = 0$$ is not a vacuum model. It is well-suited to describe matter with no pressure, which we usually call dust, but there is no motivation (at least that I know of) to treat it as if it was vacuum. The cosmological models I'm mentioning do assume though that there is a fluid filling the entire Universe with $$\rho > 0$$ and $$p = 0$$, but this is taken to be an approximation (just like water is made of molecules but we treat it as a continuous fluid, the Universe is filled with galaxies but in large scale we treat it as a continuous fluid).

• Thank you for your answer! I do not understand why $\omega=0$ isn't a vacuum model. Then, our universe with $\Lambda\neq 0$ isn't a vacuum model either, even if I would take all matter out of it, is it? Nov 25, 2021 at 7:30
• @BarrierRemoval It isn't a vacuum model because it behaves like matter does. It has pretty much the properties you would expect of dust, in the sense it has energy, but no pressure. I mean, I guess you can call it vacuum, but that would have the properties of matter. $\Lambda \neq 0$, on the other hand, does not behave like any sort of usual matter we consider in other parts of Physics, so it pretty much could only amount to modelling the properties of what we call vacuum Nov 25, 2021 at 7:50
• It could be though that there is a paper somewhere modelling vacuum with $w = 0$, but I never heard of anything like that and I can't see any reason for why we would call that vacuum. (cold) dark matter, for example, is modeled with $w = 0$ and we do not understand it as a sort of vacuum energy, but rather as a different sort of matter, because it gravitates like matter Nov 25, 2021 at 7:55