Which one is more fundamental in nature: matter or radiation? I am following a geometric perspective on abelian gauge theory as done in the lecture notes by Timo Weigand, chapter 6, pp 165-167, here:
http://www.thphys.uni-heidelberg.de/~weigand/QFT1-13-14/SkriptQFT1.pdf
He basically shows that a local symmetry of the matter field described by Dirac equation will directly give as a consequence the existence of the gauge boson field, here photon. In other words, the matter field and its geometry, imply the existence of the local vector field $A_{\mu}(x)$ that has its own dynamics.
The math makes sense to me, perhaps except for the part that we "construct" an object called the Wilson line, $C(y,x)$, that has a nice and magically useful transformation behavior, but the question is that are we saying that the theory of electrons predicts the existence of photon? In other words, which is more fundamental in nature? the matter or radiation? Can we say that radiation has evolved from matter? Does this contradict our cosmological models that say the early universe has been radiation dominated, now matter dominated, and in the future vacuum dominated? 
 A: 
He basically shows that a local symmetry of the matter field described by Dirac equation will directly give as a consequence the existence of the gauge boson field, here photon.

This is wrong. There are perfectly well-defined theories where you have matter fields but no gauge fields. And vice-versa: we have perfectly well-defined theories of gauge fields with no matter. These two objects are conceptually independent, and the presence of one does not imply the existence of the other.
The correct statement is that the presence of matter fields provide a natural motivation for gauge fields. But the opposite statement is also true: the presence of gauge fields provides a natural motivation for matter fields, so both these objects are at the same conceptual level: neither of them implies the other, and they are both good enough in order to motivate the other. Form a logical point of view, neither of these fields is more fundamental than the other.

Can we say that radiation has evolved from matter? Does this contradict our cosmological models that say the early universe has been radiation dominated, now matter dominated, and in the future vacuum dominated?

Apple and oranges. The fact that the energy density of massless particles, of massive particles, and of the cosmological constants evolve through different powers of the red-shift parameter $z$ has nothing to do with one being more fundamental than the rest.
A: 
''He basically shows that a local symmetry of the matter field described
  by Dirac equation will directly give as a consequence the existence of
  the gauge boson field, here photon.''

But it is the local symmetry, not the matter field, that is responsible for the existence of the gauge boson field. Indeed, the local symmetries of a quantum field theory are in a 1-1 correspondence with the gauge boson fields. 
Thus it is the assumption of local symmetry that forces the photon, not the presence of matter. There are lots of (perturbatively defined, nonrenormalizable) relativistic quantum field theories involving matter but no radiation.
In nature, both matter and radiation are fundamental.
A: This is a Chicken and Egg problem.
Geometrically, it is more natural to say that Gauge bosons (radiation) are consequence of having Gauge symmetry in fermions (matter). Gauge bosons play a role of connection in a vector bundle formed by fermions and their transformation properties. In fact, the boson kinetic term in Gauge theories is none less than the product of the curvature tensors!
More over, there is a simple reason for having Gauge-invariant matter: the local phase in QM must have no effect on the measurements since it disappears from the probability. This is another explanation of starting from the fermions and getting bosons from the gauge symmetry.

QFT has very profound geometrical aspects. Although, it is a quite a big detour, I'd recommend taking a look at some Differential Geometry textbooks. My favorite is "The Geometry of Physics: An Introduction" by T. Frankel.
A: In your very question, when you describe the Dirac equation, you automatically imply the answer that the most fundamental concept in nature is mathematics. It is not. Mathematical models can be very good, but mathematics is simply logic limited by the initial assumptions and the nature always goes beyond any model. So if your question is limited to some particular model like the Dirac equation or QFT, then your question isn't truly about the nature and you already have several answers like that. However, if your question is about the deep properties of the nature beyond any specific mathematical logic, then you must use physical intuition. It is not guaranteed to be always correct, but it is the only instrument of kniwledge that goes beyond the limitations of mathematics.
Another limitation that you must overcome to get a proper answer is the limitation of the "conventional science". Not every theory accepted by the "consensus" is always correct. One example from history is, "the Earth is flat". People were burned alive for trying to prove otherwise. The "consensus" is a very powerful limitation.
With all this out of the way things may suddenly become more clear. By "radiation" you seem to mean specifically the electromagnetic radiation. Although other interactions may follow the same trend. The electromagnetic radiation is emitted and absorbed by electrically charged particles. They can exist without radiation, but radiation cannot exist without them (see more on this below). So the answer in the scope of your question is that matter is more fundamental than radiation and it was matter that produced radiation in the first place. In fact, it is the same conclusion that you came to by analyzing the Dirac equation, so this conclusion does not even need to go beyond this particular mathematical model. Does this answer follow the current cosmological model? Well, It doesn't really matter. You cannot comply with every "accepted" theory while asking a fundamental question. Plus the current cosmological model is work in progress that has problems much bigger than this one. So let me summarize.

are we saying that the theory of electrons predicts the existence of photon?

Absolutely. If it didn't, we'd need a theory that does, because photons are emitted and absorbed only by charged particles.

In other words, which is more fundamental in nature? the matter or radiation?

Matter. Although the question is a bit moot, because both are the integral parts of existence, which ultimately would not be possible without both.

Can we say that radiation has evolved from matter? 

Yes. Electromagnetic radiation is emitted only by charged particles. The opposite would only be possible via the two photon physics that has not been observed and brings more problems than solutions. For example, photons don't experience time and cannot decay without hitting a real charged particle (same issue as that the neutrino oscillations are impossible with massless neutrinos). Until actually observed, the two-photon interaction is unproven. Another concern is that the matter-antimatter disbalabce is much easier to reconcile conceptually at the moment of creation than later on.

Does this contradict our cosmological models that say the early universe has been radiation dominated

Not necessarily. Matter of the very early universe could have emitted enough radiation for the early universe to be radiation dominated. In either case, the current cosmological model has much bigger problems to worry about and is hardly a high standard for fundamental questions. 
Finally, is this answer absolutely correct? No, of course not. Like anything else it is limited by our knowledge and understanding that constantly evolves. 
