How to understand critical density?

Question

In Cosmology, critical density is given by setting $\Lambda = 0$ and $k = 0$, in other words, a universe without dark energy and zero curvature. According to my understanding and Wikipedia, this critical density represents the density to keep universe stable and stop expansion. A bigger density will lead to expansion, and a smaller density will lead to contraction. However in Introduction to Cosmology, it says that this critical density represents the density for the universe to stay flat, or $k = 0$, a bigger value will give a positive curvature, a smaller value will give a negative curvature.

I got two questions about this:

1. Is the statement in Wikipedia and Introduction to Cosmology the same thing or one of them are wrong?
2. If one neglect the existence of dark energy, wouldn't the universe contract because of gravity of matter pulling it together? The main reason of the Cosmological Constant is to exert a force that stops the expansion of the universe, so how would a universe with matter somehow not contract but remain stable?

Answer 1

1: The critical density is the required density for a flat universe, so it requires only $\Omega_{\rm K}$ to be $0$, but not $\Omega_{\Lambda}$, so it also holds with dark energy if you include its density in the critical density.

If you treat it as a purely geometric term without adding its pressure and density to the energy tensor as Einstein did in his famous biggest blunder it doesn't work though, you have to use the dark energy and not the cosmological constant interpretation.

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2: Even without dark energy the density of the remaining radiation and matter might add up to less than the critical density, which means the Hubble parameter $\rm H$ would be higher than required for a flat universe of that density, which leads to negative curvature, which means a positive $\Omega_{\rm K}$, i.e. a hyperbolic universe.

If the density is higher than the critical density, $\rm H$ is too low and requires a closed universe where the straightest possible distance from you to yourself is $\rm D=2\pi c/\sqrt{-\Omega_{\rm K} H^2}$. Such a universe would stop expanding and collapse one day.

If our universe had our current Hubble parameter and matter/radiation densities, but no dark energy, that would require a hyperbolic geometry in order to satisfy relativity. So with or without dark energy, a universe with our parameters will expand forever.

With dark energy even a closed universe can expand forever and exponentially though, since all the other Omegas will approach $0$ while the $\Omega_{\Lambda}$ in the $\rm H=H_0 \sqrt{...+\Omega_{\Lambda}}$ remains, so $\rm H$ would also converge to a constant value as it does in our accelerated universe, given the then negative $\Omega_{\rm K}$ under the square root which shrinks with $\rm a^{-2}$ is not so large that the collapse with negative $\rm H$ takes over before $\Omega_{\Lambda}$ starts to dominate.