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I would like to know how much of the Big Bang theory is dependent on the constancy of the speed of light.

P.S.: It might be guessed that I am asking this because of the recent CERN news. Yes, of course, I am. I was having some chat over this topic and I hit a wall before I could come to an answer for this because of my not-more-than-school-level-Physics knowledge.

Thank you.

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Most of the questions people ask we've been closing as duplicates because frankly they are all asking the same thing ;-) But this one seems more focused than the others, in that it focuses on big bang cosmology, an aspect which hasn't been discussed in detail in other answers, so I think this can be left open. – David Z Sep 24 '11 at 18:31
@David Zaslavsky I got the impression that a random number generator is used to determine which seemingly similar questions are closed and which ones are left open and even upvoted... I am obviously on the unlucky side ;-/ ... – Dilaton Sep 24 '11 at 19:47
@Dilaton: :-P But seriously: we left the first two questions of this sort open, one for a "popular" explanation and one for a more technical analysis. The rest of the questions got closed as duplicates because they seemed to be asking the exact same thing as one (or both) of those two. However, if a question distinguished itself from the original two in some significant way (like this one, by focusing on BB cosmology specifically), we left it open. – David Z Sep 24 '11 at 19:57
(cont.) That being said, I'm certainly willing to admit that we may have been a little hasty. For one thing, the different moderators didn't really have time to coordinate and agree exactly what needed to be closed and what didn't; although I was using the reasoning described above, perhaps dmckee wasn't. We can certainly take another look at your question. – David Z Sep 24 '11 at 19:59
@David Zaslavsky Thanks, that would be fair :-). I dont think that asking about the BB theory is much more focused than asking about quantum qravity in the this context. If needed I could reformulate my question to focus it a bit further ... – Dilaton Sep 24 '11 at 20:11

The constancy of the speed of light is not really an assumption, because the speed of light is dimensional, any variation can be absorbed in a redefinition of the meter/second. The only relevant question is whether relativity is true, and I will assume you are asking "what are the consequences on the big bang if special relativity doesn't work". Further, I assume you mean that special relativity doesn't work in a certain way, namely that there is a preferred slicing of space-time which defines an absolute frame of reference for velocities.

The beginning is still dense

I think that it is safe to say that the back-extrapolation to show that the big bang is dense is safe for modifications of special relativity, because the same extrapolation works in Newtonian gravity cosmology. The scale at which cosmological models break down will be pushed back to the point where the density of matter is large enough to notice the relativity violations, which is probably before the BBN era, closer to what is now called the inflation era.

The initial singularity depends on relativity sensitively

If you don't have relativity the initial singularity which is guaranteed by Hawking's theorem will disappear almost surely. The Hawking theorem requires a strong form of energy inequality which is not even true within relativity when you have scalar fields. The requirement is that (three times) the pressure is always absolutely less than the energy. This is true for any non-scalar field, and even true for scalar fields when they don't have a vacuum expectation values. The only thing that violates it is a scalar VEV. With relativity violations, everything will violate this at large energies, and you can have no singularity.

Inflation will not work

The inflationary era is at a high enough energy density that if you modify relativity, you will have to rework the whole inflationary epoch. Inflationary theory is about as predictive as big-bang nucleosynthesis, so I think it is now an established part of science.

Why this is premature

You are going by a press-release about neutrino speed that has zero chance to be correct. The supernova 1987a neutrions plus the inaccuracies in the distance and time measurements make it certain that this new result is garbage.

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