# Two kinds of flatness? cosmological constant?

• Khashishi
In summary, the cosmological constant is a real value, a feature of space-time, that ensures the topology of the universe can't change.

#### Khashishi

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.

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Khashishi said:
For a flat universe, the average stress energy is 0, right?
The Friedman solutions are not vacuum solutions. They have an energy-momentum tensor corresponding to a perfect fluid.

Khashishi said:
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.

You are confusing local metric geometry with topology. The former does not fully constrain the latter. The EFEs determine the evolution of the local geometry not the topology.

But if I assume homogeneity, the local geometry is the same everywhere and gives the topology, right?

Khashishi said:
But if I assume homogeneity, the local geometry is the same everywhere and gives the topology, right?
No. Homogeneity enables you to extend local geometric information to the global geometry of the manifold, but it does not generally tell you about the topology. For example, a torus is flat, as is the Euclidean plane, but each have different topology.

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Khashishi said:
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?
The cosmological constant isn't chosen. It's a real value, a feature of space-time. The reason why the observed flatness is a problem is because through most of the history of our universe, the expansion has acted to amplify the average curvature (pedantically, this was the case when the dominant energy density in our universe was matter or radiation). Right now, for example, the amount of spatial curvature (that we haven't yet measured) is about 500 times what it was when the CMB was emitted. Go earlier still, and that number gets much, much larger.

If we were to just start a universe with random initial conditions, we would expect to start with a large curvature. Instead, our universe had to begin with a vanishingly-tiny curvature.

bapowell, I understand your point. But if I also assume isotropy with homogeneity, will the local curvature totally constrain the topology?

Chalnoth, I guess it depends on how to define random initial conditions. It seems weird to say that the chance of having such a flat universe is tiny when we don't have any way of developing a prior probability. If the current value of Omega is not exactly 1, then I agree that the initial conditions are fairly unlikely. But if the current value of Omega is exactly 1, then I don't see any reason to believe it is unlikely. Maybe the universe just HAD to be flat.

Khashishi said:
bapowell, I understand your point. But if I also assume isotropy with homogeneity, will the local curvature totally constrain the topology?

Chalnoth, I guess it depends on how to define random initial conditions. It seems weird to say that the chance of having such a flat universe is tiny when we don't have any way of developing a prior probability. If the current value of Omega is not exactly 1, then I agree that the initial conditions are fairly unlikely. But if the current value of Omega is exactly 1, then I don't see any reason to believe it is unlikely. Maybe the universe just HAD to be flat.
The natural expectation of a prior probability is that we would naively expect numbers to be ~1 in natural units. So it would make sense if, at the moment our universe began, the curvature was 0.2 or 0.8. But 0.0000000000000000000000001 should seem rather absurd.

What this means, essentially, is that there has to be an interesting physical process which produces the small curvature. Inflation has been proposed as a solution to this problem.

Chalnoth said:
The natural expectation of a prior probability is that we would naively expect numbers to be ~1 in natural units. So it would make sense if, at the moment our universe began, the curvature was 0.2 or 0.8. But 0.0000000000000000000000001 should seem rather absurd.

What this means, essentially, is that there has to be an interesting physical process which produces the small curvature. Inflation has been proposed as a solution to this problem.
To add a bit to this, another way of looking at it is just to ignore the initial conditions, and consider the universe we observe. It is impossible for the spatial curvature of our universe to be a very large negative number or a very large positive number: a large negative number, and our universe would have recollapsed on itself long ago. A large positive number, and it would be expanding too fast for any galaxies to form. So given that, we could potentially exist with a curvature somewhere roughly in the range of -1 to 1 (caveat: this is very ad-hoc, so I could be quite far off, but it's not going to be horribly different from this).

Instead, we have a curvature closer to zero than 0.01. That's a little odd on its surface, and should have an explanation somewhere in the dynamics of the early universe.

## 1. What are the two kinds of flatness in cosmology?

The two kinds of flatness in cosmology refer to the flatness of the universe and the flatness of spacetime. The flatness of the universe pertains to its overall geometry, while the flatness of spacetime relates to its curvature and how it affects the motion of objects.

## 2. What is the cosmological constant?

The cosmological constant is a term in Einstein's theory of general relativity that represents the energy density of the vacuum of space. It is often denoted by the Greek letter lambda (Λ) and can have a positive or negative value, affecting the expansion rate of the universe.

## 3. How does the cosmological constant affect the flatness of the universe?

The cosmological constant has a direct impact on the flatness of the universe. A positive value of Λ results in a positively curved universe, while a negative value results in a negatively curved universe. If the value of Λ is precisely zero, the universe is considered "flat."

## 4. How does the cosmological constant relate to dark energy?

The cosmological constant is often associated with dark energy, which is the mysterious force responsible for the accelerating expansion of the universe. It is believed that dark energy may be related to the energy density of the vacuum represented by the cosmological constant.

## 5. Why is the cosmological constant important in modern cosmology?

The cosmological constant plays a crucial role in modern cosmology as it helps explain the observed expansion of the universe. It also helps to reconcile Einstein's theory of general relativity with observations of the universe's large-scale structure. Additionally, it is a key factor in determining the ultimate fate of the universe.