The relation between the speed of light and the Big Bang Theory 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.
 A: 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.
