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There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions regarding this:

1) Does the fact that a manifold with a) asymptotic (space) curvature zero and b) local inhomogeneities with positive (space) curvature imply that there will be regions with negative (space) curvature?

2) a Region of negative (space) curvature implies dark energy in that region?

3) assuming answer to both 1) and 2) are true: does this represent an independent confirmation of dark energy? or there is somehow an geometric relationship relating asymptotic flatness to accelerated expansion (the traditional reason to introduce dark energy in the first place)?

EDITED: to reflect distinction between space and space-time curvatures.

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You have to be careful to distinguish between curvature of space and curvature of spacetime. When we say that the Universe is flat on large scales, we're talking about space -- that is, about a slice through spacetime at constant cosmic time. With respect to spatial curvature, statement 1 is correct: we do have zero curvature on average, and positive curvature in some regions, which implies negative curvature in other regions.

But statement 2 doesn't follow from statement 1, because in this case we want to talk about spacetime curvature. To be specific, ordinary matter produces positive spacetime curvature (i.e. a positive Ricci scalar), and dark energy produces negative spacetime curvature. But spatial curvature and spacetime curvature are different things.

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great answer. thanks i didn't realise that the paper was relevant only to space curvature – lurscher Feb 1 '11 at 21:54
Ted, you said: "positive curvature in some regions, which implies negative curvature in other regions".. Are you implying that the density of dark energy is larger in certain regions? – dbrane Feb 1 '11 at 22:16
@dbrane -- No. Once again: the statements about dark energy are about spacetime curvature, while the statements about positive and negative curvature are about spatial curvature. And anyway, the fact that curvature varies from place to place doesn't imply that the density of dark energy varies from place to place -- at most, it implies that the total density varies from place to place. DE could be uniform, with the variations caused by the other stuff. – Ted Bunn Feb 1 '11 at 22:18
@dbrane, the first question doesn't mention dark energy at all, 1st question is entirely about geometry – lurscher Feb 1 '11 at 22:31
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The universe is flat spatially, but the space is being stretched with time on the Hubble frame. This means how the space is embedded in spacetime is such that there is curvature, such as a “time-time” Ricci curvature $R^{tt}$. This solution to the Einstein field equations is such that the pressure is equal to the negative of the energy density of the vacuum. So dark energy, which is associated with this pressure is due to a positive energy density. The Hamiltonian for this is $H~=~\Lambda x^2/6$, which is similar to the spring potential. However, the force acts in the same direction as the displacement.

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thanks. Please, can you rephrase your question as answers to each specific subquestion? – lurscher Feb 1 '11 at 21:42
My main comment is on how dark energy is not due to negative energy. That is clearly not the case. – Lawrence B. Crowell Feb 1 '11 at 21:54
and why negative energy is relevant to the question? are you saying that negative energy is required on regions with negative (space-time) curvature/Ricci scalar? – lurscher Feb 1 '11 at 22:08
Negative energy suffers from a range of problems, where here I am thinking of $T^{00}$. Dark energy is not a case of negative energy. That is the main thrust of what I wrote. – Lawrence B. Crowell Feb 1 '11 at 22:34
thats correct. But i never mentioned negative energy in the question, that is why i asked why you implied that it was relevant – lurscher Feb 1 '11 at 22:41

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