October 12, 2003

Nature Fumbles the Football

A while back, I noted that some people are taking seriously that the WMAP data on fluctuations in the CMBR show “too little” power in the low multipoles ($l=2,3$). As an explanation. they’ve advocated the idea that the universe has nontrivial spatial topology.

Most likely, this is just foreground junk, and once the foreground subtraction is done better, these two datapoints will cease to stand out, and the apparent lack of isotropy at large angular scales will go away.

If the data are real, the original paper suggested the most obvious possible explanation: replace a spatially-flat, simply-connected, universe, $\mathbb{R}^3$ with $\mathbb{R}^2\times S^1$ (or $T^3$ — it doesn’t much matter), with the size of the $S^1$ somewhat smaller than the current horizon.

There are, of course, other possibilities for nontrivial spatial topology. Some will be ruled out because they would predict deviations from the observed spectrum at smaller angular scales ($l\geq 4$), others because they would predict too strong a departure from $\Omega_0=1$. But, surely, these two data points (which may go away in any case) are too little information to decide between the remaining possibilities.

So, imagine my surprise when Sonia pointed out to me that Nature’s cover story for the October 9 edition was Universe Could Be Football Shaped, flogging a paper advocating the “Poincaré Dodecahedral Space” as the model for the spatial topology of the Universe.

Since the paper provides neither a coherent explanation of why they focus on this particular space, nor a decent explication of what the space is, I thought I’d provide a public service and fill in the blanks. En passant, I hope we can forestall a flood of copycat papers, each one focussing on the authors’ favourite choice of spatial topology.

The FRW universes have spatial slices, $M$, which are either $H_3= SL(2,\mathbb{C})/SU(2)$, $\mathbb{R}^3$ or $S^3=SU(2)$. The corresponding isometry groups of $M$ are $SL(2,\mathbb{C})$, $ISO(3)$ and $SO(4)=SU(2)\times SU(2)/\mathbb{Z}_2$. These spaces are said to be homogeneous and isotropic, which is to say that, given a point $p\in M$, the six-dimensional space of infinitesimal isometries can be broken up into three “rotations” about the point $p$ and three “translations,” which move the point $p$.

Our universe looks locally FRW, but it could just as well be $M/\Gamma$ for $\Gamma$ some freely-acting discrete subgroup of the isometry group. If we want $M/\Gamma$ to be homogeneous, but not isotropic, we should demand that $g\Gamma g^{-1}=\Gamma$, for $g\in G$, a 3-dimensional subgroup of the isometry group of $M$, so that the latter acts transitively on $M/\Gamma$. It’s not really clear how physical that assumption is. Locally, the space is the same as before; it’s the global topology that may or may not admit the three global killing vectors corresponding to “translations.” If (for argument’s sake) the “obstruction” lay outside our current horizon, there’s no way we could know that our Universe was not homogeneous.

Artificial as the restriction may be, it eliminates the possibilities based on $H_3$.

Even among quotients of $\mathbb{R}^3$, there are lots of possibilities. I’ve already mentioned $T^3$, where of course, the “shape” of the $T^3$ is up for grabs. But there are other, more exotic possibilities. Here’s my favourite: the dihedral group,

(1)$D=\mathbb{Z}_2 ⋋ (\mathbb{Z}_2\times\mathbb{Z}_2)=\langle a,b \mid a^2=b^4=1, a b a = b^3\rangle$

acts freely on $T^3$ via

(2)\array{\arrayopts{\colalign{right center left} \equalcols{false}} a(x)&=&(-x_1, -x_2 +\textstyle{\frac{3L_2}{4}}, x_3 + \textstyle{\frac{L_3}{2}})\\ b(x)&=&(x_1+\textstyle{\frac{L_1}{4}},-x_2+\textstyle{\frac{L_2}{4}}, -x_3) }

So we could just as well have a universe whose spatial slices are $T^3/D$.

For quotients of $S^3=SU(2)$, we can take $\Gamma$ to be any finite subgroup of $SU(2)$ acting (say) on the right, while the $SU(2)$ group of “translations” acts on the left. These finite subgroups have a famous ADE classification. In addition to the two infinite series, $\mathbb{Z}_{n+1}$ and $Q_{n-2}=\langle a,b,c\mid a^2=b^2=c^{n-2}=a b c = -1\rangle$, there are the binary tetrahedral, binary octahedral and binary icosahedral groups (the double covers of the corresponding rotational symmetry groups of the respective Platonic solids). The binary icosahedral group, $I$, has 120 elements and — you guessed it — the “Poincaré Dodecahedral Space” is none other than $S^3/I$.

Now, it’s true that out of the infinite number of finite subgroups of $SU(2)$, most are unsuitable, as they will produce deviations at large multipoles as strongly as they will at small multipoles. Small subgroups, like $\mathbb{Z}_2$ and $\mathbb{Z}_3$ may be problematic because, to get the desired deviation in $l=2,3$, one may need the radius of curvature of $S^3$ to be small enough that the $\Omega_0$ must differ appreciably from 1.

It may well be that $I$ is the unique possibility for a group by which to quotient $S^3$, which is both large enought to allows $\Omega_0$ to be very close to 1 ($\Omega_0\sim 1.013$ is their fit to the data), while still affecting predominantly the low ($l=2,3$) multipoles. But you’d never be able to tell that from their analysis.

Posted by distler at October 12, 2003 4:26 PM

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Re: Nature Fumbles the Football

What exactly do you mean by “foreground junk”? The defining problem with the low multipole data is cosmic variance; they correspond to big patches of sky and we only have one sky to measure, right? I don’t think the WMAP data can ever be that conclusive about the low multipoles, one way or another, since the error associated with cosmic variance is so bad there.

Posted by: Bob McNees on October 15, 2003 8:37 PM | Permalink | Reply to this

Re: Nature Fumbles the Football

Read the original paper on this subject. Cosmic variance makes the error bars on the low multipoles large, but there are also serious issues with foreground subtractions. The subject of the paper was mainly how to do the foreground subtractions correctly. The “nontrivial spatial topology” aspect was a single paragraph (if I recall correctly) at the end of the paper.

Posted by: Jacques Distler on October 15, 2003 8:43 PM | Permalink | Reply to this

Re: Nature Fumbles the Football

Just to be clear, you often see plots of $C_l= a_l^2 /(2l+1)$ versus $l$. When plotted this way, the errors in the $C_l$ due to cosmic variance are $l$-independent.

If you believe the systematics, the deficit in the quadropole and octopole is something like a $2\sigma$-effect (or less, depending on whose systematics we’re talking about).

But, in a different sense, you are right. Cosmic variance is not a problem we can solve by “taking more data.”

Posted by: Jacques Distler on October 15, 2003 10:32 PM | Permalink | Reply to this

Re: Nature Fumbles the Football

It strongly looks like a paper designed to get the maximum publicity. Not only (as you pointed out) did the authors not say why this particular manifold was chosen (was it a coincidence that it happens to be one which can be described in a media-friendly way?), they were also in contact with a group (Cornish et al.) who were searching the WMAP data for matched circles.

This would be the ‘smoking gun’ for a finite, small, non-simply-connected universe. Cornish et al. announced the results the day after the Nature paper: nada. Not a solitary circle in sight. This may be a record for a theoretical paper - ruled out the day after it’s published. (I originally thought the July paper by de Oliveira-Costa et al. had already ruled it out, but it turns out they didn’t look for the 36 degree twist in the circular correlation coming from the identification of opposite faces.)

Since the authors would have known something of the progress of this search, it makes me wonder why they and Nature didn’t want to wait a week longer until the results came out that would support or rule out the dodecahedral space. It looks like they wanted the publicity more than they wanted the right answer.

Posted by: Thomas Dent on October 20, 2003 9:15 AM | Permalink | Reply to this

Circles

I’d like to understand this better. The “twist” in the circular correlations surely depends on exactly which non-simply connected space we are looking at (i.e. on the choice of group Γ). Is there a model-independent check one can make, or does one have to re-analyze the data for each choice of Γ?

Posted by: Jacques Distler on October 20, 2003 9:23 AM | Permalink | Reply to this

Re: Nature Fumbles the Football

Absolutely, the parameters are the size of the space and the discrete group. You imagine spheres of radius cT centred on us and our images in the covering space, where T is the time since last scattering. The circles are the intersections of the sphere with its image. Their relative orientation and the ‘twist’ in the correlation depend on the symmetry element that takes us into the image in question.

But they’re always correlated pairs of circles of the same radius, so in this sense it’s model-independent.

For rectangular tori (which is what the July paper looked at) the circles are obviously diametrically opposed and the correlation is ‘untwisted’. 36 degrees comes from the axial rotation necessary to take one dodecahedral cell into the next.

Cornish, et al. have a paper in 1998 with a very nice explanation. The upshot is that nearly all manifolds people had come up with (and I’m aware this is a non-rigorous statement) have circles that are either diametrically opposed or a few degrees off exactly opposite. Cornish’s most recent paper reports on the search over a 6-parameter space (radius, orientation of 1st and 2nd circles, twist angle).

Possibly the football model can squeeze through if they tweak it to make the circles small enough to escape the current data. We should know very soon.

Posted by: Thomas Dent on October 20, 2003 11:48 AM | Permalink | Reply to this

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