In the context of FRW cosmology, there is no difference in the rate of time between the epochs of the evolution of the universe. You can see that from the form of the line element
$$ds^2=-dt^2+a(t)^2\gamma_{ij}dx^idx^j.$$
That is a result of the symmetries that you assume for the matter distribution (homogeneous, isotropic) and the choice of observers that you make. So the observers that follow the expansion of the Universe, which are the galaxies more or less, perceive the same time wherever and whenever they are. The cosmological time is the proper time of all the comoving observers, as it is evident from the line element.
In the case of a Schwarzschild metric and static observers
$$ds^2=-(1-\frac{2M}{r})dt^2+(1-\frac{2M}{r})^{-1}dr^2+r^2d\Omega^2,$$
it is the factor in front of dt that makes the difference and you have different time rates for observers at different positions.
There is one more point. Someone mentions the redshift and the perceived difference of the rate of time for faraway objects. That would appear to contradict what I am saying, but it isn't. The redshift effect is an observer symmetric effect. Like in the case of SR where you have two inertial observers with different velocities and each of them thinks that the others time runs slower, when both of them actually experience proper time. That is very different from the case of the static observers near a gravitating object, where there is no such symmetry. The clock of the observer that is at bigger r runs faster than the clock of the one that is at smaller r.
This seems to be a common misconception about the big bang.
At present our theories can only suggest what happened AFTER the "bang". We cannot formulate what occurred AT the singularity with our current knowledge of physics.
At a small neighborhood around a spacetime singularity quantum gravity becomes important and we simply have no clue at present how to deal with the general relativistic singularities (e.g. black holes as well!)
When you hear of physicists talking about the big bang it is almost always a discussion of the dynamics AFTER the singularity.
When physicists discuss what happened during the planck era or at a singularity it is entirely (very) educated guesswork within the bounds of current theory (much like how theoretical physicists like hawking have come up with convincing theories about some black hole properties).
Hope that helps!
Best Answer
The first thing to think about is what exactly we mean by a big bang. A weak version of the big bang hypothesis would simply be the statement that at some time in the past, the universe was extremely hot and dense -- as hot and dense as a nuclear explosion. A stronger statement would be that, at some point in the past, there was a singularity, which is in nontechnical terms a beginning to time itself.
We have a variety of evidence that the universe’s existence does not stretch for an unlimited time into the past. One example is that in the present-day universe, stars use up deuterium nuclei, but there are no known processes that could replenish their supply. We therefore expect that the abundance of deuterium in the universe should decrease over time. If the universe had existed for an infinite time, we would expect that all its deuterium would have been lost, and yet we observe that deuterium does exist in stars and in the interstellar medium.
We also observe that the universe is expanding. There are singularity theorems such as the Hawking singularity theorem and the Borde-Guth-Vilenkin singularity theorem ( http://arxiv.org/abs/gr-qc/0110012 ) that tell us that, given present conditions, there must be a singularity in the past. These theorems depend on general relativity (GR), which at this point is a well tested, fundamental theory of physics with little viable competition. Although there are competing theories, such as scalar-tensor theories, observations constrain them to make very nearly the same predictions as GR under a broad range of conditions.
There is a little bit of wiggle room here because the Hawking singularity theorem requires a type of assumption called an energy condition (specifically, the strong energy condition or the null energy condition), and BGV is more of a model-dependent argument having to do with inflationary spacetimes (which violate an energy condition during the inflationary epoch). An energy condition is basically a description of the behavior of matter, sort of roughly saying that it has positive mass and exerts positive pressure.
Dark energy violates the standard energy conditions. So if dark energy is strong enough, you can evade the existence of a big bang singularity. You can get a "big bounce" instead. However, we have three different methods of measuring dark energy (supernovae, CMB, and BAO), and these constrain it to be too weak, by about a factor of two, to produce a big bounce. The figure below shows the cosmological parameters of our universe, after Perlmutter, 1998, arxiv.org/abs/astro-ph/9812133, and Kowalski, 2008, arxiv.org/abs/0804.4142. The three shaded regions represent the 95% confidence regions for the three types of observations. If you take the intersection of the three shaded regions, I think it's pretty clear that we're just nowhere near the region of parameter space that results in a big bounce.
There are various other observations that verify predictions of the big bang model. For example, abundances of light elements are roughly in agreement with calculations of big-bang nucleosynthesis (although there are some discrepancies that are not understood). The CMB is observed to be very nearly a perfect blackbody spectrum, which is what is predicted by big bang models. This is hard to explain in models that don't include a big bang.
Historically, cosmological expansion was observed, and cosmological models were constructed that fit the expansion. There was competition between the big bang model and steady-state models. The steady-state model began to succumb to contrary evidence when Ryle and coworkers counted radio sources and found that they did not show the statistical behavior predicted by the model. The CMB was the coup the grace, and the big bang model won. When you observe the CMB, you're basically looking up in the sky and directly seeing the big bang.
Note that although in historical cosmological models, perfect symmetry was originally assumed in the form of homogeneity and isotropy, to make models easy to calculate with, these are not necessary assumptions. The singularity theorems do not assume any special symmetry. For the Hawking singularity theorem, you just need to have a positive lower bound on the local value of the Hubble constant, and that bound has to hold everywhere in the universe on some spacelike surface. Of course, we can't observe all of the universe, and you could say that our reason for believing in such a global bound is homogeneity. However, the existence of such a bound would be only a very, very weak kind of homogeneity assumption -- much weaker than the kind of symmetry assumptions made in specific models such as ΛCDM.