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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?

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j0equ1nn
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    Obligatory reading re "single dense object": Did the Big Bang happen at a point? – ACuriousMind Dec 05 '16 at 01:30
  • @ACuriousMind Okay I've looked at the post you linked. I don't think it addresses my question. The concept of a space with no center or boundary are elementary topologically speaking. My question is how we rule out that the expansion we observe could be a local phenomenon. How do we know it's not just our neck of the universe that exploded out of the explosion we are studying? How did we reach such a global solution from such local measurements? – j0equ1nn Dec 05 '16 at 03:05
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    Most of cosmology rests on the assumption that the universe is homogeneous and isotropic on large scales (and that is true for the part of the universe we can observe). It sounds as if both of your questions are essentially saying. "What if the universe isn't homogeneous and isotropic outside of what we observe?", and the "answer" to that is simply that that may well be, but we have no reason to believe so. There can't be any evidence for that about the universe beyond the observable portion because then it would be observable since there's evidence about its features! – ACuriousMind Dec 05 '16 at 03:21
  • I guess I should learn what physicists mean by these terms and how it relates to their use in math. There certainly do exist dynamical systems having both expanding and contracting regions, easily found in nature (e.g. the ocean), hence not contradictory to the laws of the universe. I am not convinced that we have ruled out something of this nature on a larger scale. – j0equ1nn Dec 05 '16 at 03:26
  • @j0equ1nn: I guess I should learn what physicists mean by these terms... I provided two links in this comment in your other thread on this very matter. – Kyle Kanos Dec 05 '16 at 03:45
  • @KyleKanos I saw those and appreciate it, but I don't think they explain what I'm lacking. In the posts they are basically presented as what we need if we want to assume our conclusions apply to the whole universe, at which point we've gone full circle. I can think of other interpretations that would make it look like things are expanding near us. For instance, geodesic flow on a hyperbolic $3$-manifold (and by the way there are infinitely many such geometric structures that are both isometric and homogeneous, if that matters). – j0equ1nn Dec 05 '16 at 04:01
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    I think they do explain what you're lacking, but it seems you are just too adamant about your own position to appreciate the answers there. I also disagree with your assessment of the answers there, it is not assume P, then Q therefore P but We find Q, which is likely explained by P – Kyle Kanos Dec 05 '16 at 13:57
  • Re the universe not being homogeneous & isotropic, there may've been just a bit of evidence for it, perhaps more recent than the posting of this question & ACuriousMind's comment on it: A void, hypothesized by Laura Mersini-Houghton, has (reportedly) been found in the CMB, and would tend to support inhomogeneity as an alternative to dark energy. – Edouard Jul 31 '21 at 13:14

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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.

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    I really appreciate this answer in light of the fact that my questions might be annoying to many physicists. I think part of my problem is that I need to understand better the difference between actual cosmology and how it's presented in popular science. People seem to think that physicists think they have a good guess about how The Universe began. Maybe it's the oversimplification that is at issue truly. But... it's not hard to find words like "proves" and "implies" on, say Wikipedia, that gives you heartburn if you're a mathematician. And I feel like extra care is in order on this topic. – j0equ1nn Dec 05 '16 at 02:54
  • Well, no TV producer who wants to keep his/her job is EVER going to suggest a popular series on representation groups, but every second producer uses that stupid image of a ball sitting in a coordinate grid to represent gravity and warped spacetime, so you have an advantage there. Most people who ask cosmology based questions can't escape those cartoons. I think physicists don't rely on proof, so much as falsification, because whereas a math proof is timeless, (apart from Godel re axioms and Cantor re infinities, ahem...) but a physics theory is not necessarily timeless and we know that. –  Dec 05 '16 at 03:11
  • I think part of my problem is that I need to understand better the difference between actual cosmology and how it's presented in popular science This, imo is a huge problem. Physics is easier than math to look dramatic, but last week I threw out 50 percent of my popular science books, (I self study), even though they were written by heavy weights, because the drama overtook the details. If they paid academics better, this dumbed down stuff would never be written : ) –  Dec 05 '16 at 03:17
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    Right on. It's interesting to bring up Cantor and Godel though because where that stuff ended up is with us having to accept incompleteness (Cohen). With this we learn that you can't necessarily prove something by disproving (what we think is) its opposite. I feel like that messes up a lot of experimental science ... maybe not in most areas but, well we're talking about "the universe!" I can live with representation groups not being on TV. But maybe TV-watchers could swallow a little more of "that question is too big and we don't know, but what we do know is..." – j0equ1nn Dec 05 '16 at 03:20
  • Well, physics is not logic. If it was as you said from Godel and Cantor it could not be complete. Physics is making the best conclusions you can from observations and rational manipulation of the theories that can explain them. It self corrects, rather than proves, but it is indeed able to show that under certain observations certain conclusions are not possible, others are , and you then predict others and you get to some consistent theory to some limits of applicability. It is not axiomatic, but it is rational. As for the universe, see the next comment – Bob Bee Dec 05 '16 at 07:20
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    As for the universe, it is not the likelihood that the rest we can't see is the same as what we observe, it is that so much of what we observe is the same (homogeneous and isotropic, seems to follow the same laws we see here going away tens of billions of light years and going back almost 13.7 billion years. We see light emitted 380,000 years after The BIG BANG, and it reinforces what we see otherwise, with the same symmetries and just enough perturbations to cause the galaxies we observe. We predict some crazy non Euclidian space, and 100 years later observe its undulations. Next comment pls – Bob Bee Dec 05 '16 at 07:26
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    It is not physicists arrogantly saying they know something, to within certain limits and with plenty caveats, it is a mathematician arrogantly saying that because he knows about cases where there are attractors and repulsors (or whatever) that he can therefore argue that what we do know means nothing. We do know that from all our observations the universe is likely flat or slightly open (with some uncertainty, I think about 1 or 2 percent), and we have not observed attractors or big bangs or whatever elsewhere. And we keep looking. Not annoying, uneducated. – Bob Bee Dec 05 '16 at 07:33
  • @BobBee: that might be a good basis for an answer – Kyle Kanos Dec 05 '16 at 11:20
  • @BobBee Bob, comments get deleted so I think if you could put them in an answer, as Kyle says, then you should. And if you want to put in a few definitions that the OP may need to follow your arguments, and extracts from pertinent links mentioned above, that might help. As I say in my two cents, I would be delighted to get another view. –  Dec 05 '16 at 11:46
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    @BobBee Where you lose me is where you say "We do know that from all our observations the universe is likely flat" etc. If we're assuming the universe is infinite, then what you're saying is that we have measurable confidence about something based on having sampled 0% of it. 13.7 light years out of infinity is still 0. We have not observed any other big bangs, but this one accounts for phenomenon up to our limits of observation, right? I also think the response would be more professional without the insults. I came here admitting ignorance of the topic and respectfully asking to be educated. – j0equ1nn Dec 05 '16 at 20:53
  • Also please forgive me if I came across as arrogant or to be overstating my own knowledge. It was not my intentioon. What I meant to emphasize is that I am trained in a branch of mathematics that has applications to cosmology, and that I am at the beginning of exploring this application. I feel that there should be a better welcome for folks entering from this perspective, and that we can contribute. For example my background on closed manifolds lead to a significant correction to the recommended reading http://physics.stackexchange.com/questions/136860/did-the-big-bang-happen-at-a-point – j0equ1nn Dec 05 '16 at 21:06
  • Could I ask all concerned to well, Count to 10, there is no need for disrespect from either side. Thanks. –  Dec 05 '16 at 21:07
  • @j0equ1nn. I personally had no problem with your post, except when you say mathematicians choose not to talk about physics maybe that could construed badly and I don't think physicists claim to know to know as much as you imply, (regarding presumptions). But I am a total amateur, so I am lucky enough to not even notice phrases like that: ). Perhaps you can see it from a full time physicists perspective. –  Dec 05 '16 at 21:26
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    @CountTo10 I will gladly rephrase that to make it less likely to offend anyone. What I really meant is that my perspective is shared by specific other mathematicians that I know, who tend to avoid talking about physics, not that mathematicians can't be physicists or something! Let me think about this while I run some errands and then try editing to a more diplomatic approach. – j0equ1nn Dec 05 '16 at 23:32
  • Don't worry about being diplomatic, not the issue. I consider presumtive using studies of X to evaluate Y on the basis of whether it makes sense in the language/methodology/approach/etc of X, w/o studying Y. I doubt any physicist gets insulted by naive views. Maybe irked if they got the Nobel. As for the universe, there are three aspects of it: boundedness (infinite or finite), curvature (open, closed or flat), and topology (simply connected, multiply, etc). Closed implies finite but flat/open does not imply infinite. Reading: start at https://en.m.wikipedia.org/wiki/Shape_of_the_universe. – Bob Bee Dec 06 '16 at 05:15
  • And I can't really post an answer because I don't know what the question is. If it is whether there was only one Big Bang, there is the MWH from infinite inflationary universes, and some other multiverse hypothesis. It is mostly speculative thinking in physics, meaning nothing proven or measured but can't disprove it either. At that point one has to decide whether it is worth one's time. So the question of multiple Big Bangs seems to be posed in that context, and YET, argued by reference to the possibilities in mathematical models. yes, plenty is possible, but scientific truth is not that easy – Bob Bee Dec 06 '16 at 05:26
  • @BobBee I recommend viewing this question as an early attempt, among many to follow, of a mathematician to understand possible applications of his field of study. I have been told often that cosmology is a natural area to apply hyperbolic 3-manifold theory, but maybe my first attempt at crossing the gap was too general because I'm mostly struck by a broader disparity between the methodologies of mathematics and physics. From my perspective, the question is direct (see title). If it warrants no answer, this surprises me but mostly seems indicative of greater need for patient communication. – j0equ1nn Dec 06 '16 at 07:20
  • No, this is not an educational site to turn mathematicians into physicists. Read books, take courses, read professional papers, and use this site to ask questions on something you have done some thinking and self research on. That's my opinion. I know there is an intersection, but similarly physicists wouldn't go to math exchange to learn differential geometry - they may have specific questions but not is the axiomatic development of differential geometry wrong because it does not account for topological spaces. I hope you get my point, it's not personal or to discourage you – Bob Bee Dec 06 '16 at 20:30