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.