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I read somewhere that both a closed and a finite flat universe would have zero total energy in General Relativity (On the Zero-Energy Universe). But the best evidence shows that the universe is flat and infinite: "We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent" (NASA).

In this video, at 1:09:59, Lawrence Krauss says: "[Inflation] tells us, remember, that in the largest scales the universe isn't flat. It's actually closed." But the whole point of his argument is that the universe is flat and, because of that, its total energy is exactly zero (e.g., from 53:32 to 54:23).

So, I'm confused. Does inflation really predict a closed universe? If so, how is that compatible with the evidence that the universe is flat?

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  • $\begingroup$ Inflation predicts that the size of the whole universe is many orders of magnitude greater than the observable part, so it could be spherical on the large scale but still appear perfectly flat to any measurement we could do. $\endgroup$ – Nathan Reed Dec 14 '14 at 6:22
  • $\begingroup$ For one thing, the arxiv document you cite does not seem to be a peer-reviewed publication. I would not give it too much credibility. The question of the global geometry of the universe is tied to the question if general relativity is the correct theory. Personally I would not put too much belief in that, either. Since everything else on this matter derives from there, my personal suggestion would be to treat this as a completely open question, which will probably not get resolved sufficiently in this century. $\endgroup$ – CuriousOne Dec 14 '14 at 6:26
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In this link, the contradiction is "explained":

The tremendous expansion greatly dilutes any initial curvature. Think, for example, of standing on a basketball. It would be obvious that you are standing on a (2-dimensional) curved surface. Now imagine expanding the basketball to the size of the Earth. As you stand on it now, it will appear to be flat (even though it is actually curved if you could see it from large enough distance). The same idea extended to 4-dimensional spacetime accounts for the present flatness (lack of curvature) in the spacetime of the Universe out to the greatest distances that we can see, just as the Earth looks approximately flat out to our horizon. In fact, the inflationary theory predicts unequivocally that the Universe should globally be exactly flat, and therefore that the average density of the Universe should be exactly equal to the closure density. It is this prediction that we alluded to earlier when we said that there were theoretical reasons to believe that the density of the Universe was exactly equal to the critical closure density.

It hand waves that flatness is a very good approximation in the inflationary model, curvature unmeasurable in the observable universe. The curving is sent outside of it.

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  • $\begingroup$ not to sound like I'm belittling anything you said, but wouldn't this type of explanation be pretty unscientific, at least in the sense that it's essentially stating that the hypothesis is not testable. How would we go about testing to see if they're right? $\endgroup$ – Skyler Dec 14 '14 at 11:27
  • $\begingroup$ @Skyler All these are speculative models, not really testable except by consistencies with general laws and known physics. $\endgroup$ – anna v Dec 14 '14 at 11:36
  • $\begingroup$ So it's actually closed if you consider the whole universe, but "flat" in the sense that you can't really measure $\Omega-1$ in the observable universe? Did I get it right? $\endgroup$ – Wood Dec 14 '14 at 12:00
  • $\begingroup$ That is what I get from the context, that what we can measure says "flat". The analogy of the earth and the basketball. It is curvature that is unmeasurable within our accuracies in the observable universe. $\endgroup$ – anna v Dec 14 '14 at 12:14

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