Avtar:
Your comments suggest to me that you lack an understanding of the basic concepts of cosmic time and cosmic expansion. A few quotations from Eddington's Expanding Universe might help you to better understand expansion:
`The lesson of humility has so often been brought home to us in astronomy that we almost automatically adopt the view that our own galaxy is not specially distinguished---not more important in the scheme of nature than the millions of other island galaxies...
`When the collected data as to radial velocities and distances [of these galaxies] are examined a very interesting feature is revealed. The velocities are large, generally very much larger than ordinary stellar velocities. The more distant nebulae have the bigger velocities... The most striking feature is that the galaxies are almost unanimously running away from us...
`The unanimity with which the galaxies are running away looks almost as though they had a pointed aversion to us. We wonder why we should be shunned as though our system were a plague spot in the universe. But that is too hasty an inference, and there is really no reason to think that the animus is especially directed against our galaxy. If this lecture room were to expand to twice its present size, the seats all separating from each other in proportion, you would notice that everyone had moved away from you. Your neighbour who was 2 feet away is now 4 feet away; the man over yonder who was 40 feet away is now 80 feet away. It is not *you* they are avoiding; everyone is having the same experience...'
The description given by the RW metric is of an isotropic and homogeneous (i.e., maximally symmetric) three-dimensional universe that multiplies in cosmic time. When the scale-factor that describes the form of this multiplication of maximally symmetric space is equal to zero, space is singular---i.e. it has zero extent.
Please consider Fig. 1 in my essay, which graphs the evolution of Weyl's `de Sitter cosmology': at $t=-\infty$, all worldlines converge at a point; however, in the first instant of time space is infinitely large, with comoving geodesics distributed throughout Euclidean space that exponentially expands in cosmic time (Eq. (1)).
Just about everything you've said in point 1 is incorrect. The standard model assumes maximally symmetric space that expands in cosmic time, which is, moreover, the proper time of all fundamental observers. Those *are" the basic axioms of standard cosmology, so to say that it only assumes maximal symmetry of space, or that maximal symmetry and cosmic time are mutually exclusive, is just wrong.
On 2, there's no centre as you've written, just as there's no centre to the surface of a sphere or an infinite plane. When we look at distant galaxies, we are perceiving them as they were at earlier epochs, due to the finite speed of light. We don't take observations of isotropy to imply that we're at the centre of the universe, since we assume homogeneity as well, according to the cosmological principle.
On point 3, t=0 is a big bang singularity in any model where the scale-factor, a(t=0)=0. In others, like the de Sitter cosmology mentioned above, it may be different. Since the scale-factor is a *scale*-factor though, the definition is arbitrary. What matters is the model that's been empirically constrained, which tells us that the big bang occurred 13.7 billion years ago, and how it has expanded since then. It takes two points to set a scale. The rest of what you've said in this point has been discussed a lot by other people, and I'd prefer not to get into it with you.
On 4: see what I've written about point 3. Then, as described by the standard LambdaCDM model, which is a very successful empirical model, the big bang is at t=0---i.e. 13.7 billion years before now---on the clocks of *every* fundamental observer in maximally symmetric space. t=0 is not just an absolute time, though: the coordinate t is the common time held by each of these fundamental observers.
Daryl
