# asymptotic curvature of the universe and correlation with local curvature

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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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.
 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