What is the universe expanding into?
This is a very good question which is not at all easy to give a satisfactory answer to! The first time I tried to write an answer to this, we got so many follow-up questions from people who were still confused that I decided to try to answer it again, this time much more comprehensively. The long explanation is below. However, if you just want a short answer, I'll say this: if the universe is infinitely big, then the answer is simply that it isn't expanding into anything; instead, what is happening is that every region of the universe, every distance between every pair of galaxies, is being "stretched", but the overall size of the universe was infinitely big to begin with and continues to remain infinitely big as time goes on, so the universe's size doesn't change, and therefore it doesn't expand into anything. If, on the other hand, the universe has a finite size, then it may be legitimate to claim that there is something "outside of the universe" that the universe is expanding into. However, because we are, by definition, stuck within the space that makes up our universe and have no way to observe anything outside of it, this ceases to be a question that can be answered scientifically. So the answer in that case is that we really don't know what, if anything, the universe is expanding into.
Now, for those of you who want a more comprehensive discussion:
Let me begin by saying that "expanding" isn't really the best word to describe what is happening to the universe, although that is the word that is often used - a word choice which I think leads to a lot of unnecessary confusion regarding what is already a difficult topic! A more accurate word for what the universe is doing might be "stretching".
The difference between "expanding" and "stretching", for me at least, is that an "expanding universe" conjures up an image where there is a bunch of galaxies floating through space, all of which started at some center point and are now moving away from that point at very fast speeds. Therefore, the collection of galaxies (which we call the "universe") is expanding, and it is certainly fair to ask what it is expanding into.
The current theories of the universe, however, tell us that this is not the picture we should have in mind at all. Instead, the galaxies are in some sense stationary - they do not move through space the way that a ball moves through the air. The galaxies simply sit there. However, as time goes on, the space between the galaxies "stretches", sort of like what happens when you take a sheet of rubber and pull at it on both ends. Although the galaxies haven't moved through space at all, they get farther away from each other as time goes on because the space in between them has been stretched.
Of course, when we think of space in everyday life, we don't think of it as something which is capable of stretching. Space, to us, just seems like something which is there, and which everything else in the universe exists within. But according to Einstein's theory of general relativity, space isn't really as simple as our common sense tells us. If we want to understand the actual way that the universe functions, we need to find some way to incorporate Einstein's ideas into our mental picture and imagine space as a more complicated entity which is capable of doing things like "bending" and "stretching".
To help us imagine this, a lot of people have come up with analogies for the universe in which space is represented by something more tangible. For example, there is the analogy with a sheet of rubber (or sometimes a balloon) that I mentioned above. My favorite analogy, though, involves imagining the universe as a gigantic blob of dough. Embedded in the dough are a bunch of raisins, spread throughout. The dough represents space, and the raisins represent the galaxies. (To the best of my knowledge, this analogy was originally proposed by Martin Gardner in his 1962 book Relativity for the Million.) We have no idea how big the dough is at this point - all we know is that it is very big, and we, sitting on some raisin somewhere inside it, are so far away from the "edge" that the edge can't possibly have any effect on us or on what we see.
Now, someone puts the dough in the oven and it begins to expand. The raisins move apart from each other, but relative to the dough they don't move at all - the same particles of dough that start off near a particular raisin will always be next to that raisin. That is what I meant when I said that the galaxies aren't really moving through space as the universe expands - here, the raisins aren't moving through the dough, but the distance between the raisins is still getting larger.
This new picture of the universe which I am asking you to imagine is, on a practical level, much different from the old picture in which the galaxies are all moving through space away from some point at the center. A lot of concepts and definitions that seem simple to us in the old picture are much more complicated now. For example:
What is the distance between two galaxies? In the old picture, this is an easy question to answer theoretically (though not necessarily in practice!). Just get yourself a giant tape measure and clip it to a faraway galaxy, then come back to our galaxy and hold on tight. As the galaxy moves away, it will pull on the tape measure, and you will easily be able to read off the distance as the tape measure unwinds... one billion light-years, one and half billion light-years, two billion light-years, etc.
In our new picture of the universe, however, with the raisins and the dough, the tape measure will not unwind at all as the universe expands, because the galaxies are not actually moving with respect to each other! Instead, it will read one billion light-years the whole time. You could be perfectly justified in saying that the distance between the galaxies has not changed as time goes on. When you bring the tape measure back in, however, you will notice something unusual; due to the stretching of space, your tape measure will have stretched as well, and if you compare it to an identical tape measure which you had sitting in your pocket the entire time, you will see that all the tick marks on it are twice as far apart as they used to be. Using the tape measure from your pocket as a reference, you would now say that the galaxy is two billion light-years away, even though the first tape measure said it was one billion light-years away. As you can see, the concept of "distance" in this new picture of the universe is somewhat more complicated than in the old picture! It is unclear whether the universe as a whole is really "expanding" - all that we really measure is a stretching of the space between each pair of galaxies. (Note that we might have to have an "imaginary" tape measure whose atoms aren't actually being held together by intermolecular forces in order for the scenario described above to actually take place as described.)
(By the way, this analogy of the tape measure is pretty similar to what actually happens to light when it travels between galaxies. When light is emitted from one galaxy and travels through space to another galaxy, during its trip through space it also will be stretched, causing it to have a longer wavelength and therefore causing its color to appear more towards the red end of the spectrum. This is what leads us to see redshifted light when we look at faraway galaxies, and it is measurements of this redshift that allow us to estimate the distances to these galaxies.)
Where is the center of the universe? In the old picture, it is easy to say where the center of the universe is - it's the point in space that all the galaxies are moving away from. In the new picture, though, this isn't so clear. Remember, the galaxies aren't actually moving away from each other - they're sitting still! Let's go back to the dough analogy - sure, you can imagine that even if the dough is really really big, it has some point within it which is the geometric center. But this definition is not very useful. Since the dough represents the space that we live in, we have no way to see "outside" of the dough to get a sense of the entire shape and figure out where the center is. So if you are stuck inside the dough, and have no way to see anything except the dough, and if you are so far from the "edge" of the dough that you can't see it and it can't have any effect on you, then what difference do you notice between the point where you're at and the point that is actually at the geometric center of the entire blob of dough? The answer is that there is no difference, absolutely none. The concept of the "center of the universe" loses all meaning, so we don't even think about it.
In fact, we can go a step further and imagine that the center isn't even there at all! How? Well, what if instead of just being really really big, the dough were infinitely big - that is, you could walk forever in a straight line and never reach a place where the dough ends. In that case, there really would be no center of the universe - the only way you can define the center is to mark out the edges and find the point that's equally in between all of them. So if the universe is infinitely big and has no edges, then it also has no center, not even on a theoretical level.
What does the universe expand into? Finally, we can return to the original question. In our old picture of the universe, the answer would be simple, although very unsatisfying. The collection of galaxies that make up the universe is moving through space; therefore, the universe is expanding into even more space than it already encompassed. In our new picture, though, the galaxies are just raisins spread throughout the dough - their presence is largely irrelevant to the question of the universe's expansion. What we really care about is the dough, and whether or not it has a boundary.
If the dough does have a boundary, then it is legitimate to ask what is beyond the boundary that the dough expands "into". But for our universe, that is a very complicated question to ask! The boundary at the edge of the dough represents the "edge" of space. By definition, we exist within space and have no way to leave it! So we don't think there is any way to observe or measure what is beyond, unless it had some effect on us that we currently don't know about. It would be really weird to imagine reaching the "end" of space. What would it look like, for example? These are questions that we have no way to give a scientific answer to, so the simple answer is that we don't know! All we do know is that based on our current understanding of theoretical cosmology, the universe does not have a boundary - it is either infinite or it wraps around itself in some way. Observations seem to agree with these predictions in the sense that if the universe does have a boundary, we know that the boundary is so far away from us that we can't currently see it and it doesn't have any effect on us.
If the universe is indeed infinite, then the simple answer to the original question is that the universe doesn't have anything to expand into. Thinking about infinity is always complicated, but a good analogy can be made with simple math. Imagine you have a list of numbers: 1,2,3,etc., all the way up to infinity. Then you multiply every number in this list by 2, so that you now have 2,4,6,etc., all the way up to infinity. The distance between adjacent number in your list has "stretched" (it is now 2 instead of 1), but can you really say that the total extent of all your numbers has "expanded"? You started off with numbers that went up to infinity, and you finished with numbers that went up to infinity. So the total size is the same! If these numbers represent the distances between galaxies in an infinite universe, then it is a good analogy for why the universe does not necessarily expand even though it stretches.
Finally, I should point out that not everything in the universe is "stretching" or "expanding" in the way that the spaces between faraway galaxies stretch. For example, you and I aren't expanding, the Earth isn't expanding, the sun isn't expanding, even the entire Milky Way galaxy isn't expanding. That's because on these relatively small scales, the effect of the universe's stretching is completely overwhelmed by other forces (i.e. the galaxy's gravity, the sun's gravity, the Earth's gravity, and the atomic forces which hold people's bodies together). It is only when we look across far enough distances in the universe that the effect of the universe's stretching becomes noticeable above the effects of local gravity and other forces which tend to hold things together. (That is why, in the analogy of the tape measure I discussed above, the tape measure that you keep in your pocket does not get stretched, while the one that goes between two galaxies does get stretched. I bet some people were wondering about that!)
