It's a simple undeniable fact of everyday life that we live in a 3-dimensional manifold. Space around us is inherently experienced as (homeomorphic) to R3. There's clearly no possibility we'll experience anything else ("having" more or less dimensions suddenly, or experiencing a point with no surrounding open set). Of course, this just shows we live in some 3-manifold. But which one? The tacit assumption is usually that it's R3. Physicists regularly posit this (spacetime just adds a differential manifold structure), or sometimes compactify space by adding a single point at infinity; strictly speaking, this makes space a 3-sphere (homeomorphic to the surface of a four-dimensional ball), but puts any differences from R3 literally infinitely distant from us.

What if that's wrong? The surface of the Earth, locally, is obviously a 2-manifold, and for a long time people believed it therefore had to be an infinite plane (R2) or, more commonly, a disk. They did this just because those are the easiest manifolds to think of, and the scale of any non-local structure (e.g. routes which circle the globe) was larger than anything they had experienced. Eratosthenes guessed the Earth must be a sphere, but it took considerably more direct proof than his to convince mankind in general. Some misguided people still believe in a Flat Earth Theory. Could our universe be some more complicated 3-dimensional manifold, with the nearest effects too far away to have been noticed yet?

3-dimensional manifolds are very varied. Here are some things they can do, with easier-to-grasp analogies from 2-dimensional manifolds. To begin with, they can be closed, just like a 2d sphere or torus.

Like a torus, they can be closed "in different ways": on a torus, you can circle just round the tube, or go at right angles to that and go all around that way too, but there's no way to deform one path continuously into the other. (When someone nodes homotopy group properly you should read that...) Just as a torus can be built by gluing a rectangle along two opposite edges, and then along the two remaining edges (think of video-game were you leave one edge to reappear on the opposite side), a 3-torus (with 3 different directions to "go around") can be built by gluing a cube along three pairs of opposite faces (you may need a 4-dimensional friend for the last stage). Think of walking along a torus, laying out some string. Eventually you get back to the end you left behind, and tie your string into a loop. There's no way to gather up the string now (unlike doing the same thing on a sphere) -- it encloses a "hole" which has no adequate description in 2d terms; the same thing could happen if our universe is a 3-torus. In a complicated manifold you could have any number of peculiar "circular" paths and relations between them.

Next, there's no reason our universe should be orientable. If you were a 2d physicist living on a Klein bottle and you sent a top spinning away from you, it would one day come back from behind you, spinning in the opposite direction. Wham goes global conservation of angular momentum, et cetera.

The manifold we live in is patently metrisable: we experience and measure distance all the time. Is the metric bounded? In a 3-sphere it would be (go far enough anywhere and you'll get back to your starting point), and in R3 it wouldn't.

Going back to those closed paths, the "hole" (in 4 or more dimensions) can extend infinitely. Consider living on an infinitely long cylinder instead of a torus (one direction is "normal", the perpendicular direction is like one on the torus). Or, with the same topology but a different metric, what if space were a little like a 2d space composed of two planes (in R3, each with a small circular hole, the two holes connected by a narrow tube. Now think multiple planes, and multiple tubes....

What effect does all this have, and can we find out the truth? Well, obviously, like the great explorers we could, in theory, find out one day by going out in a spaceship to chart the universe. Since this will take several gigayears (there don't seem to be any non-trivial closed paths in our mmediate vicinity, or anything like that....), let's label that Plan B. As mentioned above, some physics we're used to are just plain wrong in certain 3-manifolds. But remember what we had to do with that top on the Klein bottle: send it round the universe! Anything of that sort will again take way too long. So is it hopeless?

I think not. First of all, there is in fact an experiment which has already been running for us since the birth of the universe. Light! We may not be able to travel mllions of light years to chart the universe, but the light from distant galaxies has already done that for us. If we live in a 3-sphere, for instance, we might see the same ultra-distant galaxy shining at us from two different (opposite?) directions; we'd identify it by the exact spectrum, or whatever. In a 3-torus and so on we'd see it from several directions, perhaps estimating the distances too, thus beginning to piece together a map of our manifold. I don't know that anyone has tried to compare "different" galaxies, but at least there's a chance we could handle these observations.

The other "ray of light" comes from quantum mechanics. Wouldn't it be fun if the theory of the tiniest details of the universe let us gain an insight into the grandest scale of its sturcture? Objects (in quantum mechanics) don't exist at a point, but rather have a spread-out wave function; because it is fairly localised, the position and other quantities are determined reasonably well. However the wave function satisifies a differential equation with certain boundary conditions. If the boundary conditions of a free particle are that the wave function continues to infinity (under our usual assumption of living in R3), the solutions are different than if there exist non-trivial closed paths. Just as the solutions of such an equation on a circle or sphere become discrete, instead of the continuous soluions permissible on a line or plane, so we might find certain energy-states quantised just by the characteristics of the manifold we live in. A local experiment on something with a long enough wavelength (perhaps a graviton?), observed at a small enough scale, might reveal the existence (and length) of a route to circumnavigate the universe.

They're both not experiments we're likely to be able to do tomorrow, but the theoretical numbers don't seem to be completely ludicrous (unlike taking a spaceship to "go check"). I think "Project Eratosthenes" could literally be "the biggest thing" we ever learn about the universe.

There's clearly no possibility we'll experience anything else ("having" more or less dimensions suddenly, or experiencing a point with no surrounding open set")

In contrast to this established view M-theory suggests that we may be constrained to a four-manifold embedded in a higher dimensional space (constrained on the Brane). (for an early proposal see hep-th/0001113 by Akama et al.)

In such models gravity is free to propagate into the higher dimensional space.

This has observational consequences. The Einstein field equations have an extra term. It should be observable in two ways. Directly, gravitation on length scales of a few millimeters should deviate from Newtonian gravity. It has implications for the evolution of Inflation in the early Universe.

Should such effects be observed we would discover that our world is stranger than heretofore thought.

This month's (August 2000) Scientific American carries an article about it, but I have not read it and cannot vouch for it.

Thursday 9th October 2003: "Universe is shaped like a soccer ball, scientists say"

Nature vol. 425, 593-595 (2003)
Luminet et al.: http://arXiv.org/abs/astro-ph/0310253
http://news.bbc.co.uk/2/hi/science/nature/3175352.stm

Friday 10th October 2003: "Our results rule out the recently proposed dodecahedron model of Luminet, Weeks, Riazuelo, Lehoucq and Uzan."

de Oliveira-Costa et al.: http://arXiv.org/abs/astro-ph/0307282
http://www.hep.upenn.edu/~angelica/topology.html
Cornish et al.: http://arXiv.org/abs/astro-ph/0310233

What's going on?

Quick introduction

On large scales, the Universe clearly has one time and three space dimensions. In most scenarios of cosmology the time dimension is required to be an infinite or semi-infinite line or a line segment with past and future boundaries. Then the question is, what form do the spatial dimensions take at each point in time?

The properties of space, for example that it appears to be continuous and smooth, and that it is possible to measure distances, means that it should be modelled by a mathematical entity called a manifold. Specifically a 3-manifold, since this has three independent directions. 3-manifolds are rather difficult to visualize, so most of the examples are going to be 2-manifolds instead, like a 2-sphere, which the surface of a balloon more or less is.

So the question becomes, which 3-manifold do we live in?

The possibilities

If we start off without preconceived demands, there are an awful lot of manifolds we could think of. I'll summarize a few of the properties the manifold might have.

• Infinite: The manifold continues without limit in all directions and has infinite volume.
• Finite: The spatial size and volume of the manifold are finite numbers and the shortest distance between two points can never be greater than a given length.
• Closed: Roughly, the manifold is finite but has no edges. You can keep going in any direction without coming up against a boundary beyond which it is impossible to pass. For example, the 2-sphere or the 2-torus (surface of a doughnut). Of course if you keep going long enough you'll get back close to where you started.
• Finite with boundary: meaning, there is an "edge of the Universe". For example, one side of a sheet of paper. This option is usually discarded immediately as being problematic. The problem is, what physically happens at the "edge" to stop anything travelling beyond it? You'd need physics at the edge to be radically different from physics everywhere else to make this work.
• Simply connected: In a simply-connected space, any two paths from one point to another can be continuously deformed into each other. Imagine taking two pieces of elastic within the manifold, both of which are tied between the same two endpoints. Then you can always bring the two together without meeting any obstructions. Or, any closed loop can be continuously shrunk to zero size.

The 2-sphere is simply-connected, but the 2-torus is not. Imagine one piece of elastic goes one way round the donut and the other piece goes the other way: then you can't bring them together while keeping both on the surface of the torus. Or, a loop that goes around the torus can't be shrunk to zero size. Manifolds that are non-simply-connected are said to have "non-trivial topology".

Now, where could we actually be living? As I said, manifolds with boundary are usually chucked out for not being sensible. Based on the observation that the Universe appears to be homogeneous and isotropic, we might restrict ourselves to manifolds of constant curvature. The simplest such manifolds are called Einstein spaces, since old Albert himself used them as models of the Universe. However, this still leaves a certain degree of choice.

For a flat Universe, we could have infinite flat space, or one of many possible 3-tori: the 3 dimensions of the torus and the way it connects back up to itself are free parameters.1 For a positively curved space, we could have a 3-sphere or one of many other possibilities with different topology. (The manifold has to be finite in this case.) For negative curvature, there is the infinite hyperbolic geometry, and yet more closed spaces -- in fact, the number of negatively curved spaces is much, much larger than the number of flat or positively curved ones.

How big is it?

For a finite space or a space with curvature, the size and radius of curvature of the space must be greater than a certain distance. This is just so that the currently observable Universe can fit inside. For example, if we were living in a small 3-torus, we'd observe periodicity, i.e. we'd see the same stars and galaxies repeated in different directions. We don't, at least so far, so the size has to be of the order of a few billion light years or larger. Interestingly, for any given closed curved manifold, there is a rigid relation between the radius of curvature R and the total volume: the volume is a constant times R^3.

Also, observations of the cosmic microwave background (CMB) by WMAP (combined with other data) tell us that the amount of spatial curvature of the Universe has to be pretty small, i.e. that R has to be fairly large. The usual way of summarizing this is to say "the Universe is flat", however this isn't quite correct. The results tell you the curvature directly, but they are given in terms of a parameter called Ω which is the ratio of the total density of the Universe today to what the density should be in a flat Universe. Ω<1 means hyperbolic geometry, Ω>1 means spherical geometry and a closed Universe. They find Ω=1.02 with a error of 0.02, so in fact the real value might easily be anywhere from 0.98 to 1.06.

How can we tell?

As just mentioned, if we live in a rather small manifold, it might be possible to see apparent replicas of our Galaxy, the Large Magellanic Cloud, etc.. However, since these would probably be several billion light years away, we'd be seeing ourselves when we were much, much younger, and in all likelihood unrecognizable. That is, if we could see ourselves at all through the dust and gravitational lensing that distorts images at large distances.

If the parameter Ω does have a value different from 1, and we can measure it sufficiently precisely to rule out a flat Universe, we will have a very big clue about the identity of our manifold. However, given the success of cosmological inflation so far, the deviation of Ω from 1 is likely to be unmeasurably tiny, and the manifold (if finite) would be much bigger than the observable Universe. For a finite manifold that's sufficiently large, we'll never be able to tell the difference between it and an infinite space. Still, the discovery of a curved Universe would be worth a Nobel Prize or two, so people will carry on looking.

Looking for Matched Circles

The most powerful method of looking for a small manifold is the same CMB radiation that tells us that Ω is nearly 1. The recent observations consist of a detailed map of the temperature of microwaves in all directions over the entire sky. These microwaves come from the Big Bang -- more precisely from an era when electrons were not bound into atoms and were free to scatter with photons and stop them travelling large distances. To put it simply, the Universe was opaque.

As the plasma expanded and cooled, suddenly the electrons no longer had the energy to keep moving around freely, and they got bound into neutral atoms. At this point the photons were free to go, and most of them have continued unmolested whizzing through space up to the present day. The point at which this happened is called the Last Scattering Surface (LSS). So we're taking a thermal imaging picture of the LSS, which is the Universe as it was roughly 13.7 billion years ago. Now, because microwaves move at the speed of light, the LSS we see comes from those bits of the Universe that live 13.7 billion light years away from our current position. What we see is (the surface of) a sphere of radius 13.7 billion light years.

Now remember that in many closed spaces, it's as if there is a replica of ourselves at a certain distance away in certain directions. That replica must also be observing the LSS, which is again a 13.7 billion ly sphere, centred on the replica. Now if the distance between us and the replica is less than 27.4 billion ly, the two spheres (actually, our sphere and its own image) must intersect in a circle. And there's not just one circle of intersection, there are many replica circles, all with the same pattern of microwave temperature along them. If we look in the direction of one point on the circle, we must see the same temperature as if we looked at the corresponding point on one of the replica circles.

So the idea is to look at pairs of circles on the sky and see if any pairs of circles have correlations in this way. Since the latest CMB data came out, the search has been going on. (There are an awful lot of circles one could choose!)

Stop Press?

Now we come to the Big Story I started with. Another effect of living in a small manifold is that there is a limit on the distance scale, effectively the wavelength, of temperature fluctuations. The pattern of temperature on the LSS can be broken down into spherical harmonics -- effectively waves of different sizes on a spherical surface. But clearly it's impossible to have a wavelength that's bigger than the size of the manifold. So, in some small manifolds, you expect a much smaller amplitude for the very largest spherical harmonics, the so-called quadrupole and octopole, than you'd get in an infinite space.

And what the WMAP results show is indeed a very low quadrupole, and an octopole which is also somewhat below expectations. So, some cosmologists and astronomers (and a mathematician, G. Weeks) got together and found a particular closed manifold, the "Poincaré dodecahedral space", which seems to fit the data. What's led to the veritable media frenzy is the fact that this space has a basic cell which is a solid dodecahedron: a Platonic solid with the same symmetries as a soccer ball.

The dodecahedron is made into a closed manifold by identifying opposite faces: if you go out of one side, you come in again the opposite side. This "wrap-around" property can also be imagined if you think of space being filled with one dodecahedron after another, each of them being a replica of the next.

But if you happen to have a few dodecahedra handy, you can verify that you can't fill space with them. If you try to put 3 dodecahedra around a point, you'll find there is space left over. To make them fit properly, the angles of a pentagon should be 120°, rather than 108°. However, if space has positive curvature, and you make the pentagons the right size, they do have angles of 120°! So the Poincaré dodecahedron requires Ω to be greater than 1: to be precise, between 1.012 and 1.014.

Spoiling the Party

The real story here, however, isn't that the Universe may be "football-shaped". It isn't. The real story is how Nature, the "world's leading scientific publication", accepted, and put on its front cover, a paper that was shown to be wrong exactly one day after its publication.

The day after the Nature publication, another team led by Angélia de Oliveira-Costa at U-Penn issued a revised version of a paper they had already put out in July of this year, about the significance of the WMAP data, in particular the low quadrupole, and reporting the result of a search for "small Universe models" (small closed manifolds). The Penn team used the matched circles technique to look for candidate manifolds that could explain the data. They found nothing. The only revision they did was to add the sentence "Our results rule out the recently proposed dodecahedron model of Luminet, Weeks, Riazuelo, Lehoucq and Uzan."

They also pointed out that the low amplitude of the quadrupole could occur by chance. The fluctuations occur according to a statistical distribution: for the small fluctuations, the sky is big enough for us to see lots of them, so we can average out and get a small statistical error. But by definition, we only see very few of the very largest fluctuations on the sky, so the possible error is much larger. It's not all that surprising that the deviation from expectations happens just where the errors are largest.

In the last few days, another team (Cornish et al.) has confirmed that there are no matched circles in the WMAP data. As good Popperians, we should rejoice when it turns out that observations rule out a particular theory. However, this doesn't explain why Nature published a theory that had apparently been ruled out three months previously and was definitively ruled out the day after publication. It looks as if the editors and referees didn't have a clue what they were doing.

Nature may be the "world's leading scientific journal" in biochemistry and one or two other fields, but for theoretical physics and cosmology it's worse than useless. Anything published in Nature on those subjects is likely to be both incorrect and over-hyped. Real theoretical physics and cosmology happen in Physical Review (Phys Rev), Physics Letters (Phys Lett), the Astrophysical Journal (Ap J) and Monthly Notices of the Royal Astronomical Society (Mon Not Roy Astron Soc).

Update, Thursday 16th October 2003: Gah. Science is even more complicated than you thought.

de Oliveira-Costa et al. revised their statement again. It now turns out that their July paper didn't rule out the dodecahedron model, because they didn't do a complete search of the possible correlated circles. The reason is fairly easy to see: opposite faces of a dodecahedron have a relative twist or rotation of 36°. So the temperature along one of the circles is correlated with the temperature along the other circle, only 36 degrees out of sync. They didn't search for this "twisted" type of correlation.

But, Cornish et al. just finished a really comprehensive search for matched circles, including all possible twists. They still found zilch. The team who proposed the dodecahedral model are trying to wriggle out of it, but it still looks to me like this is a theory that died the day after it was published.

Apparently, the two groups were exchanging information up until the Nature article was in press. This makes it even stranger: if you have a theory, and the data analysis that's going to support or disprove it is very nearly finished, why rush to print the theory before you know the result of the analysis? But this is a well-known tactic for gaining publicity. If the theory is supported, you get the bonus of having "predicted" the data; if it's disproved, you get the publicity of the original announcement and of the subsequent controversy. In the long run this doesn't do science any good, since it gives the impression that scientists are more concerned with getting publicity than with finding the right answer. Which we're not. Honest.

1. Despite what a doughnut or bagel looks like, the 2-torus and 3-torus involved in this writeup are flat manifolds. Mathematically, they're rectangular or cuboidal chunks of Euclidean space with opposite sides or faces identified. An actual doughnut is positively curved on the outside and negatively on the inside, because of the necessity of circumscribing the hole while staying in 3 dimensional space, but it's possible to endow the surface with a flat metric.

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