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According to a recent paper on the arXiv, they do. How credible is this result? The abstract says:

The detection of magnetic fields at high redshifts, and in empty intergalactic space, support the idea that cosmic magnetism has a primordial origin. Assuming that Maxwellian electromagnetism and general relativity hold, and without introducing any `new' physics, we show how the observed magnetic fields can easily survive cosmological evolution from the inflationary era in a marginally open Friedmann universe but fail to do so, by a very wide margin, in a flat or a marginally closed universe. Magnetic fields evolve very differently in open and closed Friedmann models. The existence of significant magnetic fields in the universe today, that require primordial seeding, may therefore provide strong evidence that the universe is marginally open and not marginally closed.

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I added the hyperlink to the paper. It looks like an intriguing argument. This is not an endorsement yet. ;-) –  Luboš Motl Oct 6 '12 at 16:00
Thanks! I was in the process of adding the link and abstract as you were editing also.... –  FrankH Oct 6 '12 at 16:03
@FrankH I'd be curious to here your answer... –  Hal Swyers Oct 26 '12 at 21:11
@HalSwyers I read the paper and it sounds convincing to me but I am far from an expert and there is a lot I did not fully understand so I was hoping some expert would explain whether they were convinced or could find flaws in the explanation. Inflation can still be consistent with a negatively curved open universe since the curvature can be very close to zero (which is what inflation would require) but still be non-zero. Experimentally the curvature is consistent zero with and experimental error of about 1%. So there is room for it to be slightly negatively curved. –  FrankH Oct 26 '12 at 21:43
It's actually not a recent argument. Check the references in the paper, esp [8], it dates back several years. I went and read some of that but I'm not much wiser. The thing is the following: what they call "ricci identities" are just that, identities. They are always valid, you learn nothing from this. Since they don't write down a Lagrangian, I can't tell whether or not the coupling of electromagnetism to GR is the standard coupling or not. If it's the standard coupling, it would be convincing. If not, not. –  WIMP Dec 25 '12 at 15:48

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