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This question assumes Newtonian cosmology which considers a pressureless dust model of the universe. According to Carroll & Ostlie's Introduction to Modern Astrophysics textbook, I quote directly:

$$\Omega=\frac{\Omega_0(1+z)}{1+\Omega_0z}=1+\frac{\Omega_0-1}{1+\Omega_0z}$$ where $\Omega$ is the density parameter, $\Omega_0$ is the present value of the density parameter and $z$ is the redshift parameter.

This equation shows that the sign of $\Omega-1$ does not change, and in particular that if $\Omega=1$ at any time, $\Omega=1$ at all times. The character of the universe does not change as the universe evolves, it is either always closed, always open or always flat. This equation also shows that at very early times, as $z\to\infty$, the density parameter $\Omega\to1$ regardless of today's value of $\Omega_0$. Therefore the early universe was essentially flat. The assumption of a flat universe will greatly simplify the description of the first few minutes of the universe.

I have a couple of questions regarding this paragraph

  1. Why does the equation imply that the sign of $\Omega-1$ does not change, if for instance the redshift parameter was negative, wouldn't the sign change from positive to negative?
  2. Why does this equation imply that the character of the universe does not change as it evolves, I do not qualitatively nor quantitatively understand why this is so
  3. Qualitatively, why does a positive, negative and unitary (value of 1) density parameter correspond to a closed, open and flat universe respectively
  4. I understand quantitatively that the density parameter changes the sign of k which describes the sign of the total energy of the universe, but I don't understand qualitatively why different signs of total energy will change the shape of the universe.
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  • $\begingroup$ I don't study astronomy or cosomology so I'll leave my answers here, for now at least. (1) Cosmological redshift $z$ cannot be negative because it is defined w.r.t. a stationary atom's/molecule's spectral lines. (2) The character of the universe, in this context, is determined by whether $\Omega$ is either positive (closed), negative (open), or zero (flat). $\endgroup$ May 14 at 19:06
  • $\begingroup$ (3) The density parameter is defined to be the ratio of the mass-energy density of the universe, with the critical mass-energy density of the universe. If you have more mass-energy than 'critical', it will eventually stop expanding and then recollapse (because of gravity). $\endgroup$ May 14 at 19:06
  • $\begingroup$ (4) What is $k$? And when you say 'sign', you're talking about the sign of $\Omega-1$, right? $\endgroup$ May 14 at 19:11
  • $\begingroup$ Your answers seems to make sense to me, for q4 k is actually involved in the expression for total energy, and I suspect its purely consequence of the math that k happens to define the nature of the universe. Thank you! $\endgroup$
    – Lucas Tan
    May 16 at 3:38
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  1. The value of z=1/(1+a) is always positive.

  2. Some characteristics of the universe do change.

  3. See https://en.wikipedia.org/wiki/Shape_of_the_universe.

  4. "the sign of k which describes the sign of the total energy of the universe" is wrong. The sign of k relates to the shape of the universe, not its energy.

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