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Physicists have estimated the universe to be flat and infinite, but I can't seem to understand what this actually means. First of all, what do they mean when they say that the universe is infinite, do they mean that space is unbounded? Or do they mean that the space is currently bounded and is expanding forever (so the bounds are getting further apart) ? And if space is unbounded, how does it expand, how can infinity get larger? Finally a flat universe could also be bounded if for example it has a toric shape which has zero curvature, could this be a possible shape of the universe?

What also confuses me is that, since the beginning of time, the universe had a very small size, but how did it become infinitely large in a finite amount of time?

Dimtsol
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    Consider a simple map $\mathbb{R}^3 \mapsto \mathbb{R}^3$, $\vec{r} \mapsto 2\vec{r}$. It maps an infinite space into an infinite space and yet every point gets twice as far away from a center (therefore, the space is expanding). Note that this is a simple example and doesn't model the expansion of our universe. – DrLRX Dec 08 '17 at 18:46
  • Expansion refers to the change of size, not the size itself. I don't know how a derivative would be defined in this case (if one exists) though. – jjack Dec 08 '17 at 22:54

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First of all, what do they mean when they say that the universe is infinite, do they mean that space is unbounded?

No, they mean that it's not closed. By analogy, imagine an infinite Euclidean plane as an open universe, and the surface of a sphere as a closed universe. We normally do not even try to do general relativity on a spacetime that has a boundary, in the sense of an edge like the edge of a piece of paper.

And if space is unbounded, how does it expand, how can infinity get larger?

Check out Hilbert's hotel: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

Finally a flat universe could also be bounded if for example it has a toric shape which has zero curvature, could this be a possible shape of the universe?

Here you seem to be using "bounded" in a different sense. A torus doesn't have a boundary. Yes, in theory the universe could have the topology of a torus. But so far nobody has seen any evidence that the universe has a nontrivial topology.

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    no one has seen any evidence that the universe has a trivial topology either. Einstein's equations only can give you local geometry, and not global topology. – Dr. Ikjyot Singh Kohli Dec 08 '17 at 21:31
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Something that is infinite and expanding, on the face of it, seems difficult to reconcile; when a finite object expands, such as a balloon, we quickly judge that it is expanding because it's boundary, that is the surface of the balloon, is expanding.

When the object is infinite there is no boundary so this characterisation won't do.

However, if we mark two points on the balloons surface we can see when the balloon expands the distance between the two marks is expanding too. We can take this as a definition in the case of an infinite object; that is an infinite object, like for example, the infinite plane, is expanding when between any two points the distance between them is increasing.

Mozibur Ullah
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Logic tells me that an infinite universe cannot expand. The question comes up because, our perspective being the vicinity of Earth, everything at a distance seems to be redshifted and thus appears to be moving away, giving support to the concept of an expanding universe. However, given an infinite universe featuring multiple big bang arenas, then from a perspective within any given big bang arena everything observable would appear to be moving away. There could be a potentially infinite number of such arenas in existence at any given time across infinite space, but from within an arena, the issue is a matter of how far you can see out. The observable arena is but a tiny segment in a greater universe beyond, and the finite speed of light and the life span of a big bang arena combine to hide distant arenas from our view. On a whole, local expansion is not evidence of an expanding universe, but instead it is evidence of local expansion with an observable arena.

Bogie_smiles
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    I agree that the concept of an infinite universe expanding may appear unintuitive, but there is no problem with it mathematically. If you want to claim that it's ruled out by logic, you need to explain that logic, but please note that this site is for answering questions in the framework of mainstream physics. I'm not exactly sure what you mean by "big bang arena", but it gives me the impression of big bangs occuring in a pre-existing void, which is very different to standard Big Bang theory. Please see https://physics.stackexchange.com/q/136860/123208 – PM 2Ring Jul 16 '20 at 03:58
  • I should not have posted my hypothesis which is not mainstream. My post should be deleted, but I'll clarify why I was maintaining that an infinite universe cannot expand. My personal definition of "universe" is "everything including infinite space". The reason there is no problem mathematically may be focused on the concept of a finite universe with an expanding "boundary" which some equate to the surface of an inflatable sphere with nothingness beyond. My view is that the infinite space of the universe is unbounded. – Bogie_smiles Jul 16 '20 at 13:04
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    Big Bang theory doesn't rule out the universe being finite. But even if it is finite, it doesn't have a boundary. Here's a recent question on this topic: https://physics.stackexchange.com/q/565328/123208 – PM 2Ring Jul 16 '20 at 13:35
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    Thank you for the comment and link. I have not deleted my non-mainstream comments because your clarification comments are worth noting. – Bogie_smiles Jul 16 '20 at 19:43