Flat universe density parameter

For a flat universe, density parameter Ω_universe=1. How does negative signage of a constituent density parameter, such as that of curvature index Ω_k, which can be 0,1,-1 affect the signage of Ω_universe? If Ω_k were to be converted to its energy density, which is much less than the energy density of the other constituents, and even if Ω_k = -1, would Ω_universe = -1?

If you include ##\Omega_k## in the sum of Omegas you get 1 by definition. Regardless of the curvature.

Ranku said:

How does negative signage of a constituent density parameter, such as that of curvature index Ω_k, which can be 0,1,-1 affect the signage of Ω_universe?

In addition to what was explained by Orodruin, it is worth clarifying that the "curvature density parameter" ##\Omega_k## is, by definition, equal to 1-(##\Omega_M##+##\Omega_\Lambda##+##\Omega_R##), so its value is not restricted to being 0, 1 or -1. In fact, with that definition, a negative value of ##\Omega_k## implies a positive curvature of the universe.

Orodruin said:

If you include ##\Omega_k## in the sum of Omegas you get 1 by definition. Regardless of the curvature.

Put another way, each of the ##\Omega## parameters is normalized so that the total adds up to 1. Saying that the sum of the ##\Omega## parameters is 1 is equivalent to saying that all of the components have to add up to 100% of the total.

ohwilleke said:

Saying that the sum of the ##\Omega## parameters is 1 is equivalent to saying that all of the components have to add up to 100% of the total.

It seems to me that this interpretation is not entirely correct because it would imply that, for example, a positive curvature contributes negatively to the density parameter, while dark energy, mass and radiation, added together, contribute more than 100% of the total.

Jaime Rudas said:

It seems to me that this interpretation is not entirely correct because it would imply that, for example, a positive curvature contributes negatively to the density parameter, while dark energy, mass and radiation, added together, contribute more than 100% of the total.

Negative percentages are common in quantum mechanics. Why not cosmology too?

Jaime Rudas said:

it would imply that, for example, a positive curvature contributes negatively to the density parameter

Yes, that's correct. The "density parameter" is not the same thing as actual density. It is just another way of organizing the math.

ohwilleke said:

Negative percentages are common in quantum mechanics. Why not cosmology too?

"Negative percentages" is only a workable description of things that occur in QM if you adopt an interpretation of that term that has no counterpart in a classical theory. So this response is not a valid one.

PeterDonis said:

Yes, that's correct. The "density parameter" is not the same thing as actual density. It is just another way of organizing the math.

The "curvature density parameter" ##\Omega_k## is more of an indicator of how far from flatness a space is, than a true indicator of density.

Jaime Rudas said:

The "curvature density parameter" ##\Omega_k## is more of an indicator of how far from flatness a space is, than a true indicator of density.

Yes.