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I'm confused about the relationship between two seemingly different concepts of flatness of the universe.

1. Spatial flatness. This is the lack of any curvature on a large scale. Simple enough...

2. Energy density flatness. If the energy density is higher than a critical value, then the universe will eventually contract. The normalized energy density is written as Omega. So if Omega > 1, the universe will contract. If Omega < 1, the universe will keep expanding. If Omega = 1, the universe will keep expanding but asymptotically approach 0 expansion.

The concepts seem to be connected through the Einstein Field Equations. But there's something that bothers me.

The cosmological constant seems to be able to be chosen arbitrarily to give any spatial curvature we want, given an observed average energy density. AFAIK, the cosmological constant is equivalent to choosing a zero point energy. For a flat universe, the average stress energy is 0, right?

Now I don't see what the deal is with the flatness problem. If we are choosing the cosmological constant to match our flatness of the universe, then does that give us Omega = 1 automatically? It supposed to be a problem that Omega is close to 1, but how could it be anything else but 1?

Another thing, if the universe is/was/will be spatially flat at any point, it seems it has to stay that way forever. The overall topology of the universe can't suddenly change. So maybe the universe is flat because it started out flat. No problem there, is there? It seems to me almost like the cosmological constant is a fudge thrown in there that ensures that the topology of the universe can't change. Do we have any reason to believe it is a constant and not dependent on the density? I mean, if the energy density of the universe dropped somehow (through expansion), the universe topology can't change so the cosmological constant needs to change.

1. Spatial flatness. This is the lack of any curvature on a large scale. Simple enough...

2. Energy density flatness. If the energy density is higher than a critical value, then the universe will eventually contract. The normalized energy density is written as Omega. So if Omega > 1, the universe will contract. If Omega < 1, the universe will keep expanding. If Omega = 1, the universe will keep expanding but asymptotically approach 0 expansion.

The concepts seem to be connected through the Einstein Field Equations. But there's something that bothers me.

The cosmological constant seems to be able to be chosen arbitrarily to give any spatial curvature we want, given an observed average energy density. AFAIK, the cosmological constant is equivalent to choosing a zero point energy. For a flat universe, the average stress energy is 0, right?

Now I don't see what the deal is with the flatness problem. If we are choosing the cosmological constant to match our flatness of the universe, then does that give us Omega = 1 automatically? It supposed to be a problem that Omega is close to 1, but how could it be anything else but 1?

Another thing, if the universe is/was/will be spatially flat at any point, it seems it has to stay that way forever. The overall topology of the universe can't suddenly change. So maybe the universe is flat because it started out flat. No problem there, is there? It seems to me almost like the cosmological constant is a fudge thrown in there that ensures that the topology of the universe can't change. Do we have any reason to believe it is a constant and not dependent on the density? I mean, if the energy density of the universe dropped somehow (through expansion), the universe topology can't change so the cosmological constant needs to change.

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