Is there any realistic, understandable, provable (even in some extent) explanation/model for the extent of the universe? What is its shape? and Why? I mean physical explanations not philosophical since the question is about a physical entity, the universe. When I try to think about this it is almost I cannot think at all!
Note I am almost satisfied that there is no answer for this single question! I paid attention to all comments and spent time for watching the video-lectures recommended (I don't recommend) just to seek the answer, however, I got nothing about the question at all. The discussion and lectures are mathematical games far being physical to me. I respect all the science involved in cosmology but for this question even great researcher (some got Nobel prize recently) have nothing to say except playing with formulas and graphs. I was looking for physical meaning.
I lost my interest in this question.
Answer
I have split my answer into two different sections (shape and extent). Although the two are very much related in a physical and mathematical sense, they illustrate different aspects of the cosmology of our Universe. I have allowed myself to post a rather long answer, as I believe that the questions posed here cannot be answered easily by a simple statement. Instead it is my belief that the answers have to be approached from several directions, all rooted in mathematics and physics, but nevertheless converging on a more philosophical ground. Philosophical grounds are, after all, the common starting place for some of the most exciting theories within physics.
Shape
As for the shape of the Universe, this depends on the overall curvature (which is sort of the same thing). The curvature depends on the amount of matter and energy in the Universe, through Einstein's equations. The curvature and energy content can thus be estimated using a cosmological model (the typical one is $\Lambda$CDM). The curvature effects the expansion history of the Universe, which can be probed using large-scale structures, like galaxy clusters, and the cosmic microwave background radiation (CMB). Using this cosmological model along with observations, one can calculate the likelihood of the parameters in the model. Doing this one gets a curvature equal to, or very close to, zero. The analogue of which is a flat, smooth and infinitely extended bedsheet.
The curvature then appears as a natural parameter in the geometrical model of spacetime. How you choose to interpret this parameter is entirely up to you. Many people like to think of it in terms of analogies. The shape of the Universe, or spacetime in GR, is in mathematical terms analogous to the shape of other geometrical objects, like oranges or saddles. Since the spacetime in GR is 4-dimensional, the only physical analogies we can think of will be lower-dimensional equivalents hereof. Assuming that GR is the only theory we should use on large scales, these analogies are actually very accurate. This is not necessarily the case for more general cosmological theories, some of which propose so-called topological defects (eg. cosmic strings) and other geometrical objects which are not so easily visualized.
Thinking about the shape of the Universe is thus a very interesting and highly non-trivial endeavor.
Extent
The extent of the Universe is also a very interesting and non-trivial thing. Some solutions suggest that the Universe may be infinite. Others suggest that is is finite, but with no boundary (like the 2D surface of the Earth). Now there are a few natural ingredients, which turns the discussion in an even more interesting direction. These are: the finite speed of light, the finite age of the Universe and the expansion of the Universe. You might be familiar with Olbers' paradox. If not it is highly recommended reading.
Basically it illustrates our inability to see infinitely far away in the Universe. Thereby the term 'The Observable Universe'. This is actually more a statement about causality than of technological limitation. Because the speed of light is finite, and because the Universe is expanding, there is a limit to the distance to which we can send and receive signals. We can actually still observe some very distant objects, which emitted their light a very long time ago, but we will never be able to send a signal back to them, because the Universe is expanding too quickly, so the light will never be able to catch up with it, even if it travels for an infinite amount of time. This example illustrates the core of the matter about the extent of the Universe, which is horizons. Beyond these horizons we can never hope to reach, with our current understanding of things.
Another interesting consequence of the expansion of the Universe is that far away objects are sliding out of our horizon. Because of this you could say that the amount of content (of visible matter) in the observable Universe is actually decreasing, while the Universe in itself is ever-expanding.
The reason for bringing up this matter of horizons is that making statements about the Universe beyond the horizons (and out to infinity beyond) is on the verge of philosophy. This does not mean that one should stop thinking about it, or ask questions, quite the contrary. But it means that consensus is often hard to come by. Many exotic ideas exist about the nature of the un-observable Universe. These include multiverse, eternal inflation, local void, and many others. I suggest reading about those if this is of interest.
I'd like to end on a reference which is very much on the point of horizons. It features some very informative diagrams, which illustrate some of the different cosmological horizons that are present in our Universe. I should warn that they are not easily understood, even by professionals, but they convey a lot of information and insight, if given some time.
[1] T. M. Davis, “Fundamental Aspects of the Expansion of the Universe and Cosmic Horizons,” Feb. 2004.
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