Flat Torus Universe and Matched Circles

Can a flat universe still be finite? Yes. Flatness is a local property, and a flat space can still close on itself like a video-game screen whose edges join. If such a box were smaller than the visible universe, the microwave sky would repeat the same pattern along pairs of circles. Planck found no such circles, so any box must be about as large as the visible universe or larger.

Resize a flat periodic box and watch the same sky spots return as matched circle pairs, then compare each match with chance.

Box edge $1.2\,\chi_{\mathrm{rec}}$

Matched circle pairs $9$

Largest circle radius $53.1^\circ$

Box-axis match vs chance $0.95\ \mathrm{vs}\ 0.56$

This synthetic teaching sky uses one flat cubic three-torus with box edge L, measured in units of the comoving distance to last scattering. A Gaussian random field with a Gaussian-smoothed spectrum is made exactly periodic on a grid of 64 cells per edge and read on a thin last-scattering sphere. Optional white noise is independent in each one-degree sky cell. The model omits curvature, the other compact spaces, the integrated Sachs-Wolfe and Doppler terms, the thickness of last scattering, masks, foregrounds and beams, and it uses no Planck or WMAP map. The chance level is the largest match among unrelated axes in this one sky, not a calibrated false-detection probability. The box is a teaching example, not a measured shape of our Universe.

The same sky, read along circles

Synthetic sky, not a Planck map

Whole-sky map with the predicted circles

Temperature around each circle

Circle search against chance

How the circles and the match are computed

\cos\alpha=\frac{|\mathbf g|}{2\chi_{\mathrm{rec}}},\qquad S(\phi_*)=\frac{2\langle T_A(\psi)\,T_B(\psi+\phi_*)\rangle}{\langle T_A^2+T_B^2\rangle}

Each lattice translation shorter than twice the last-scattering distance gives one pair of circles. The search compares back-to-back circles on 96 evenly spread axes and the short lattice axes, at every rotation. The grey chance level uses only blind axes more than eight degrees from any clone axis.

Cornish, Spergel and Starkman (1998), circles in the sky · Planck 2015 results XVIII, background geometry and topology

Continue with the course on the shape of space

One-sentence intuition: A copy of the observer at distance less than twice the last-scattering distance makes a pair of circles with identical temperatures. A box larger than that leaves no circles, and the sky alone cannot tell it from infinite flat space.

Core mathematical model

Copies of the observer

\mathbf x\sim\mathbf x+L\mathbf n,\qquad \mathbf n\in\mathbb Z^3,\qquad \mathbf g=L\mathbf n

Points that differ by a whole number of box edges are the same point. In the covering space, a copy of the observer sits at every lattice point. Distances are in units of the comoving distance to last scattering.

Where two spheres meet

\cos\alpha=\frac{|\mathbf g|}{2\chi_{\mathrm{rec}}},\qquad |\mathbf g|<2\chi_{\mathrm{rec}}

Our last-scattering sphere and the sphere of a nearby copy meet on a circle. We see it twice, around the direction of that copy and around the opposite direction, with the same angular radius.

Comparing two circles

S(\phi_*)=\frac{2\langle T_A(\psi)\,T_B(\psi+\phi_*)\rangle}{\langle T_A^2+T_B^2\rangle}

The statistic equals one for identical circles and scatters about zero for unrelated ones. Taking the largest value over many rotations and axes also lifts it by chance, so a match must clear the chance level of the same sky.

Common difficulties

Flat does not mean infinite

Typical misconceptionIf space is flat, it must go on forever.

Better mental modelCurvature is measured locally. A flat three-torus has zero curvature everywhere and a finite volume. Local measurements cannot tell the two apart; only light that has travelled around the space can.

Run the experiment

01
Find the repeated spot
Keep the default box and watch the two photon paths. Before looking at the data row, predict how many circle pairs a box of edge 1.2 makes.
What to observe: Light from one spot reaches you from two directions. The six nearest copies give three pairs at 53.1 degrees and the twelve diagonal copies give six pairs at 31.9 degrees.
02
Grow the box past the limit
Drag the box corner outward. Predict the edge length at which the last pair disappears, then check it in the scan chart.
What to observe: The circles shrink and vanish when the edge reaches two last-scattering distances. Beyond that, the best match found sits on the chance level everywhere.
03
Add noise
Choose Noise hides the match. Compare the box-axis match with the chance level, then save the record.
What to observe: Noise as strong as the signal roughly halves the match. The pair is still physically there, but in this sky it no longer stands above chance.

Flat Torus Universe and Matched Circles

The same sky, read along circles

Whole-sky map with the predicted circles

Temperature around each circle

Circle search against chance