This post is the second in a series of posts that met (decreasingly) with some contention from the physics community, do to my status as a pure mathematician at the very beginning of trying to discuss applications of my field of study (hyperbolic $3$-manifolds), about which I was pretty ignorant. The previous one was: Can we see enough of the universe to have a valid opinion on whether it's expanding? and the following two are Regarding the universe, doesn't "almost flat" mean "not flat?" and What are the applications of hyperbolic $3$-manifold theory to cosmology? I hope that others can appreciate my choice to evoke a potentially annoying conversation as opposed to remaining ignorant. Please be patient with me and if I say something stupid, let me know and tell me why.
When say "the universe," there is a built in problem of definition since if we really knew what it was we would be done. We often get around this by saying "the observable universe," which loosely means the portion of space which is close enough to us to be "theoretically" measurable. In fact, the very definition of observable universe includes the assumption that it is expanding because, according to the theory, that expansion is the very thing which limits the theoretical measurability.
The naive idea of the Big Bang is that, since things are drifting away from each other, we can trace back their trajectory to a single more dense object, from which they must have exploded. Given that we are only able to measure up to a certain sphere around ourselves, isn't it a bit presumptuous to say that this characterizes the entire creation of the universe? (After all we, as humans, have a long history of overestimating the scope of our observations.)
Let's grant that the observable universe 4 was the result of such an explosion (maybe we should call it "a" Big Bang). My specific question is, what evidence do we have that objects in the universe at large, beyond the observable portion, also originated from this same bang? For instance, what properties of matter can we study locally that would imply this globally, in the mathematical sense? [As I pointed out in my other post (linked at top) a homogenous dynamical system can very well have multiple repelling and attracting points and we could just be near a big repelling point.] Lastly, if we do not have reason to extrapolate such a thing, should we not caution the general community about confusing "The Universe" with the part we're studying, in this manner?