What evidence is there that The Big Bang was not just a local phenomenon? 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?
 A: I can only definitively answer your last question.

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?

We do! That caveat is found in lots of answers on this site.  Among non physicists, there is in my experience a huge amount of interest in fundamental questions regarding the universe (it's the basic human need to know, which strangly enough generally fades away when math is introduced :) The link you were referred to above must be amongst the most common referrals for certain classes of questions, such as "what is the universe expanding into?". 
This really the only  general community we can reach, apart from the rare mathematician who wanders in...

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?

How can we answer this question, except on the balance of probabilities? One possible way of checking that the hidden part is much the same as the observable part is to check for gravitational anomalies. Is one part of the observable region being affected by an inhomogeneous distribution of whatever "things" are in the hidden part, and to my extremely limited knowledge, we have not detected anything in that regard.
There is an endless of possibilities of what is beyond the observable part, and it is useless to even raise one them as an example, as they are all unprovable.
Because I  don't consider this as an answer, I  feel it is accepable to put the question back on you. Why should there be a difference in any way between the observable and the hidden part, when we do have evidence of an initial start, we do have a good idea as to why we have this division between observable and hidden, even if we have no idea as to it's actual mechanism? Yes, I appreciate that this only applies to the observable part, but that just highlights the difficulty or impossibility of giving an answer.
Finally, I should say, as it's obvious anyway,  that this simplistic, naive reply to your question is given by someone who has very limited cosmological knowledge, but I would be surprised, and also delighted, if any other answer, however sophisticated and detailed, will significantly alter anything I have written here.
