#### Re: flat = infinite ?

[Caveat: as I said elsewhere I'm not current in this – the last time I was people had no idea if the universe was flat or not. People who *are* current should correct me!]

You are correct that flatness on its own does not imply infinite (where by 'infinite' I technically mean 'infinite or topologically odd': for instance the universe could be topologically a torus, which can be flat, but we assume it is not like that). However there are some strong reasons to believe that flatness does imply infinite.

First of all, flatness *and the cosmological principle* really do imply infinite. One way of phrasing the cosmological principle is that we're not in some special place in the universe (we may be at a special *time*: what's called the perfect cosmological principle would say that we're not, but we don't think that's true). What that means is that, for everyone, the universe should look about the same, and that means that it seems to be flat for everyone, everywhere, which can't be true if it is finite.

So, well, the cosmological principle is just a principle, and although it seems extremely compelling it could be wrong. So if we assume it is flat, but the cosmological principle is wrong, then here is some kind of 'edge'. And then there's a question about what that edge would be like, which is a whole other thing which I think is problematic. But we can avoid thinking about that, because, if we believe that the universe is expanding, then, if it's finite, then once it was a lot smaller. And in fact, once upon a time it was as small as we like. Now I have to think hard about the geometry but I am almost sure that this means that, whatever we mean by the 'edge', if we look back far enough then it appears in our past light cone: we can see it, in principle. But we don't see it: the CMB is extremely isotropic for instance. Well, perhaps it's so far before the time where the CMB originated that not only is it not visible to us but that whatever trace it left has got lost.

Well, that could be, but this is all looking like special pleading: in order for this idea to work we have to make a whole series of assumptions which, if we just did not make them, would make things hugely simpler and explain what we observe just as well. So, well, let's drop those assumptions and assume it's infinite, which is the simplest case.

(You can ignore the following two paragraphs: I got sidetracked.)

So the other option is that the cosmological principle *is* true, but the universe is not, quite, flat: it has very small, positive curvature. There's a parameter, Ω, which describes the curvature of the universe, and if it's flat, then Ω = 1. But, if we're right about the universe expanding, then one thing that happens is that (1/Ω - 1) (which is 0 if Ω = 1) gets scaled *enormously* during the expansion. It gets scaled by about a factor of 10^60 since the 'Planck era' (which is very, very close to the big bang). So if Ω is very close to 1 now, it must have been *extremely* close to 1 in the early universe: in the early universe Ω must have been 1 to within 1/(10^62) or so.

This is known as the 'flatness problem', and there are various answers to it. One is the anthropic principle: we only happen in universes where Ω is close to 1. That's not very satisfactory. The traditional answer is inflation, which can be used to drive Ω to be very close to 1 even if it didn't start that way. A third answer is to say, well, the universe really is flat for some reason we don't know.

(Start reading again here.)

So, I've got a bit side-tracked here: the last two paragraphs aren't actually very relevant. I think the text before that is really the answer: if the universe is flat, but finite, then the cosmological principle is false (which is OK), and we need some kind of model of what the edge would be which we don't have, but even without such a model we would expect to see artifacts of the edge in the early universe, and we don't.

To answer your other question: yes, the expanding-balloon model works for universes with positive curvature. It's just useful to show that an object can both be finite but not have an edge in this case.

And finally note that I've probably said 'we know that ...' &c above. What that really means is that 'if a whole bunch of current theory and observation is correct, then ...'. All of those things could be wrong. We just don't currently think they are (the paper this is all about might, of course, be a hint that they, or some of them, are wrong!).