How to read Planck's cosmic topology limits

Every statement below carries one of four labels. A published result is in the refereed Planck paper or in the 2004 WMAP letter. A preprint result is in the COMPACT paper as read in version 4. A model prediction follows from the geometry of the cubic three-torus, as derived in this course. An open problem is something nobody has settled.

Section 4.2: what the search computes

Planck uses the circle statistic of Cornish, Spergel and Starkman in the form optimised for small-scale structure by Cornish and collaborators in 2004. Each circle is Fourier transformed and the modes are weighted so that fine structure counts more than broad gradients. The statistic still equals 1 for a perfect match and has mean 0 for unrelated circles.

Two versions are computed. The statistic labelled with a plus sign compares points ordered the same way around both circles, called phased. The one with a minus sign reverses the order along one circle, called anti-phased. The paper states that orientable topologies produce anti-phased matches, so for the cubic torus the minus-sign curves are the ones to watch.

The search is limited to back-to-back circles, pairs centred on opposite points of the sky. That covers the cubic three-torus, whose pairs are all back to back. The threshold comes from 300 Monte Carlo simulations of the SMICA map, processed like the data, and is set so that fewer than 1 percent of simulations would yield a false event. Temperature and polarization are calibrated separately. Label: published result.

A common misreading is to treat the dotted threshold line in the later figures as a theoretical curve. It is an empirical level from those 300 simulated skies with no topology.

Section 4.4.1 and figure 6: the smallest circle

Figure 6 is a simulation, not data. It shows the anti-phased statistic against circle radius for a simulated polarization map of a cubic three-torus with edge $2H_0^{-1}$, with noise like that of the SMICA map. The thick curves rise to peaks at the radii of the true circle pairs. The dotted line is again the false detection level.

Where the expected peaks drop below the dotted line marks the smallest circle the search can find. Planck extrapolates the height of the peak at 18 degrees towards smaller radii and finds that the minimum radius lies between 10 and about 15 degrees. To be conservative it takes $\alpha_{\min} \simeq 15^\circ$. Label: published result.

The misreading to avoid is taking the peaks in figure 6 as evidence for topology. They are the injected signal of a simulated universe that was built to be a torus.

Figures 11 and 12: the data

Figure 11 shows the temperature maps and figure 12 the E-mode polarization maps. In each, the upper panel plots the anti-phased statistic and the lower panel the phased statistic, against circle radius from 0 to 90 degrees. Four curves correspond to the four Planck component separation methods, Commander, NILC, SEVEM and SMICA. The dotted line is the 1 percent false detection level.

No curve rises above the dotted line at any radius larger than $\alpha_{\min}$. That absence is the result. The only peak is at 90 degrees, and the caption explains it: a circle of radius 90 degrees centred on a point is the same great circle as the one centred on the opposite point, so it matches itself. It is a property of the search, not a signal.

Polarization uses 40 percent of the sky and temperature 74 percent, after masking. Planck notes that the polarization limits are nonetheless at least as strong as those from temperature. Label: published result.

Section 5.1.1: from an angle to a box size

The geometry chapter showed that the largest circles of a cubic torus come from the six nearest images, with $\cos\alpha = L/2\chi_{\mathrm{rec}}$. If no circle larger than $\alpha_{\min}$ exists, then

Here $R_i$ is the radius of the largest sphere that fits inside the fundamental domain, half the edge for a cube. Planck states this at 99 percent confidence and also as $L \gtrsim 6.0\,H_0^{-1}$, using $\chi_{\mathrm{rec}} \simeq 3.1\,H_0^{-1}$.

Check the arithmetic yourself. $\cos 15^\circ = 0.966$, so $L \gtrsim 1.93\,\chi_{\mathrm{rec}}$, and $1.93 \times 3.1 \approx 6.0$. The earlier analysis of the 2013 temperature maps gave $0.94\,\chi_{\mathrm{rec}}$. Model prediction plus published result: the conversion uses the cubic geometry, the angle comes from the paper.

The bound holds with a stated condition: the paper assumes that the orientation of the fundamental domain relative to the mask allows the circles to be detected. It also applies to topologies that predict back-to-back circles, in a flat universe that is otherwise the standard cosmological model.

Section 5.1.2: a second, independent method

Section 5.1.2 fits the large-angle temperature and polarization maps directly with the correlation pattern a torus would imprint. For the cubic torus the marginal likelihood requires $L > 6H_0^{-1}$, again $R_i > 0.97\,\chi_{\mathrm{rec}}$. The profile likelihood gives a weaker $R_i > 0.79\,\chi_{\mathrm{rec}}$. For a slab, a space that wraps in only one direction, the limit is $L > 3.5H_0^{-1}$, or $R_i > 0.56\,\chi_{\mathrm{rec}}$. The abstract states these limits at a log-likelihood ratio of $-5$ relative to the best-fit simply connected flat model. Label: published result.

The two methods use different information, circles at moderate angular scales and correlations at the largest scales, and land on the same number for the cube. The abstract calls them numerically equivalent.

What the COMPACT preprint adds

The COMPACT collaboration's paper, arXiv:2211.02603v4 of 27 November 2024, starts from the null circle searches by WMAP and Planck. It summarises them as showing that no closed path through our location around the universe is shorter than 98.5 percent of the diameter of the last-scattering sphere. Its footnote 6 states the condition more exactly: the searches found no excess of matched pairs at 99.7 or 95 percent confidence for clone separations up to that value. Label: preprint result, built on published searches.

For the cubic box, 98.5 percent of the diameter corresponds to circles of radius $\arccos 0.985 \approx 9.9^\circ$, close to the lower end of Planck's estimated range for $\alpha_{\min}$. That arithmetic is ours. We did not open the earlier searches that COMPACT cites, so this guide does not claim which of them sets the figure.

The paper then makes its main point. In the plain torus every observer sees the same thing, so the bound on the shortest loop through us is a bound on the box. In spaces whose faces are glued with a rotation, the length of the shortest loop depends on where you stand. The shortest loop in the space can avoid us, and the paper finds it can be shorter than the diameter of the last-scattering sphere by a factor ranging from 2 to at least 6 without violating the circle searches. Label: preprint result.

The paper also notes that the cube or any other fundamental domain should not be read as the shape of the universe. The physics lies in which points are identified. And it proposes off-diagonal correlations of spherical harmonic coefficients as a probe of the manifolds that circle searches cannot test. Label: open problem.

What the limits leave open

Three questions remain open. Whether our universe is finite at all is unanswered: a box larger than the visible sphere produces no circles. Whether a twisted flat space with a loop shorter than the visible diameter hides from the circle searches is allowed by the COMPACT analysis. We read that analysis as an arXiv preprint and did not check its journal status. And whether other statistics can reach beyond the circle limit is a subject of current work.

A checklist for the next topology paper

1. Find which topologies the search covers, for example back-to-back circles only.
2. Note the orientation convention of the statistic and which one the topology predicts.
3. Read how the false detection level was set: number of simulations, what they include, what fraction is allowed.
4. Locate the smallest detectable circle and how it was estimated.
5. Redo the conversion from angle to length with the stated distance to last scattering.
6. Separate the bound on the shortest loop through us from a bound on every loop in the space.
7. Record the version of each source and whether it is refereed.

Sources: Planck 2015 results XVIII, arXiv:1502.01593v2, COMPACT collaboration, Cosmic topology Part I, arXiv:2211.02603v4, Cornish, Spergel and Starkman 1998, arXiv:astro-ph/9801212v1, Cornish et al. 2004, Physical Review Letters 92, 201302, arXiv:astro-ph/0310233v1.

Open the near-the-limit preset and compare its 15 degree circles with the Planck threshold.