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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
@LubošMotl - please add an answer! I really would like to hear your (now) considered opinion. I would accept your answer so the current non-answer does not get the bonus... –  FrankH Oct 26 '12 at 17:54
...The non-answer got deleted, but I still want to hear @LubošMotl 's opinon –  FrankH Oct 26 '12 at 19:00
@FrankH I'd be curious to here your answer... –  Hal Swyers Oct 26 '12 at 21:11
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