A reader asks:
This seems to be as good a place as any to ask a question about closed universes.
See, in a lot of popular science books, they teach you that an "open" universe is one where space is infinite, saddle-shaped, and keeps expanding forever; a "flat" universe is infinite, plane-shaped, and the rate of expansion eventually peters out to zero; and a "closed" universe is finite, sphere-shaped, and eventually contracts in a big crunch. They then talk about the cosmological constant and "dark energy," which make our universe expand at an accelerating rate, something that doesn't fit the taxonomy of possibilities for the universe's topology, and which they do not relate back to that taxonomy in any way.
Can a universe with lots of dark energy be a closed universe? Will a closed universe with dark energy keep on expanding and accelerating, or will it eventually collapse in a big crunch like a "normal" closed universe? Is the three-type Taxonomy only relevant given certain energy conditions? (Strong/weak/null)
Oh, and I almost forgot:
are there any good reasons to think that the universe is closed in the first place, other than Kalam-esqe arguments against actual infinities?
It sounds like these books were just adding the new material about the cosmological constant to the old discussions without doing the hard work of going back and revising it so that it makes sense.
The Bad Old Days
In the old days (pre circa 1998) people didn't know about the acceleration of the universe, and they thought that the universe just consisted of ordinary radiation and matter (where for these purposes, dark matter is a form of matter). In the old days, the model of closed, flat, and open works exactly as you say: a closed universe (spherical geometry) will recollapse, and open one (hyperbolic geometry) will trend to a constant rate of expansion (in terms of distance / time) and a flat one is right on the edge and will expand forever at a slower and slower rate (but still getting arbitrarily large).
Given the rate of expansion, it takes a certain amount of energy density to get a flat universe. Too much, and you get a sphere, too little and you get hyperbolic space. (The expansion or contraction of the universe makes it hyperbolic in the absence of matter.) These are the 3 kinds of geometries which are homogeneous (the same everywhere) and isotropic (the same in every direction). On average, the observable universe seems to be homogenous and isotropic, so it's got to be one of these three (a.k.a. an "FRW cosmology").
However, this was confusing for several reasons. One is that the cosmological data kept suggesting that there wasn't enough energy in matter to get anywhere close to a flat universe, yet other data seemed more consistent with a flat universe. A flat universe is also a natural consequence of inflation since it stretches out the pre-existing geometry to exponentially large distance scales. Also, the universe seemed like it wasn't quite old enough to explain all the structures in it.
Now we know that there is an additional form of energy which is confusingly called "dark energy" (but I dislike this name, because it makes people think it has something to do with "dark matter".) Most likely it is just a cosmological constant, a constant energy density in all of space.
Now it turns out that for purposes of determining the spatial geometry, a positive cosmological constant counts positively (so it helps to close the universe). But when you calculate its effect on the expansion of the universe, it counts negatively, as repulsive gravity.
This may seem like odd behavior because energy and mass are equivalent, and we all know that mass causes gravitational attraction, not repulsion. But in turns out that in General Relativity, both energy density (associated with time) and pressure (associated with space) lead to attractive gravity. Negative pressure is called tension, and tension therefore causes antigravity.
In ordinary matter travelling at low speeds, the amount of pressure/tension is typically very small compared to the energy density. Radiation which travels near the speed of light has a lot of pressure, but that only makes gravity stronger.
On the other hand, a positive cosmological constant has tension equal to its energy density. Something has tension if, when you stretch it out, it's energy increases. But the energy of the cosmological constant is proportional to the volume, so when the volume increases the energy increases proportionally. Hence the tension in each spatial direction is equal to the energy. Since there are 3 dimensions of space and only 1 of time, the antigravity due to the tension is 3 times larger than the gravity due to the energy density. Hence the antigravity wins! So paradoxically, the gravitational effects of this tension just make the universe want to grow faster! Unlike the usual effects of tension, which cause things to shrink in on themselves.
On the other hand, if the cosmological constant were negative (it isn't, but suppose) its effects would be reversed: it would make the spatial geometry more hyperbolic, but would decelerate the expansion.
So, once you include a cosmological constant, the rules change (as you guessed). You can still have the same 3 types of spatial geometry (the words "open", "flat", and "closed" describe the spatial geometry, not the dynamics). But with a positive cosmological constant, even a universe with closed topology can sometimes expand forever, if it gets big enough for the cosmological constant to take over. (Matter thins out, while the CC doesn't, so when the universe is small the matter is more important, and when it gets larger the CC is more important.) On the other hand, with a negative cosmological constant, even an open cosmology will always eventually recollapse when it gets big enough.
(The various energy conditions you mention place limits on the allowed energy density and/or tension/pressure, so not surprisingly these have certain implications for what a cosmology can do. Note that a positive CC violates the strong energy condition—which allows for a bounce, at least in the case of a closed universe. While a negative CC violates the weak energy condition, which requires that any FRW cosmology which is neither expanding nor contracting at some time, must be closed. (OK, technically it also allows space to be flat, but only if the matter energy is exactly 0, which is unrealistic.))
Our universe seems to have a positive cosmological constant, which fixes all of the problems I mentioned above. The cosmological constant seems to give us exactly the extra energy density we need to get a flat universe. Yet it also causes the universe to be currently accelerating in its expansion (lengthening the projected time back to the Big Bang); this acceleration has been confirmed by surveys of supernovae in the past. So everything seems to hang together consistently.
As far as we can tell from current observation, the universe is exactly flat (with experimental error of about 1-2% over scales comparable to the observable universe) However, a flat geometry is right on the knife's edge between the spherical and hyperbolic cases, so actually this is perfectly compatible with the universe having a tiny positive or negative curvature, as long as the radius of curvature is big enough.
So really it could still be any of the three cases, or else something more irregular. As I said, inflation blows up the size of the universe, so regardless of the initial geometry, the observable universe will look flat after enough inflation. Outside the observable universe, for all we know, it could be some other shape, perhaps it isn't even symmetrical.
There is really no particularly good physics reason, apart from aesthetics and philosophical bias to think that the universe should be closed or open. I personally don't think much of the "Kalam" argument that actual infinities are impossible, but I do find it distasteful that in an infinite homogeneous universe everything (including all possible histories of the Earth) would happen infinitely many times in different places.
Also on the speculative hypothesis that the universe originated from some kind of quantum fluctuation, or no-boundary condition, I think one expects it to be closed. But this kind of thing is extremely speculative.
If I had to place a bet with a metaphysical bookie, my money would be on closed (but enormously large so that we could never tell). But this is my own personal guess, not a conclusion of Science!
(Incidentally, even if the topology of space is flat or hyperbolic, it would still be possible for the universe to be finite in size and therefore closed, so long as it has nontrivial topology. For example, space could be a really big "torus" where if you go far enough in one direction, you come back around on the other side, like in some video games. Locally, such a universe couldn't be distinguished from the infinite case, but globally it would be different. Astronomers have done measurements looking for nontrivial topology in the sky. They haven't seen anything, but of course they wouldn't if it happened on a scale much bigger than the observable universe!)
On the other hand, if the universe really does have a positive cosmological constant than (regardless of its spatial geometry) the final outcome seems secure. If we extrapolate the current laws of physics to the far future (assuming no changes or interventions), we get an exponentially growing universe. The matter thins out and becomes unimportant, and you end up with a very tiny final temperature (corresponding to the analogue of Hawking temperature but for cosmological horizons instead of black hole event horizons).