How can an infinite universe expand?

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?

• 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 8 '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 8 '17 at 22:54

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.

• 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 8 '17 at 21:31

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.