Can a flat universe be finite?

A screen whose edges join

Think of an old arcade game in which a ship that flies off the right edge of the screen comes back in from the left, and one that leaves the top reappears at the bottom. Nothing on the screen is curved. A ship flying in a straight line keeps flying in a straight line. Yet the screen has no edge the ship can hit, and its area is finite.

Mathematicians call this a flat torus. The word torus also names the surface of a doughnut, but the doughnut you can hold is bent. The flat torus is defined only by the rule that the right edge is the left edge and the top edge is the bottom edge.

Now do the same in three dimensions. Take a cube and declare that each face is the same place as the opposite face. Leave through the ceiling and you enter through the floor. Leave through the east wall and you enter through the west wall. This space is called a three-torus. Every small region inside it is ordinary flat space, and the whole of it has the finite volume of one cube.

Why flatness alone cannot decide

Telescopes can measure how space bends by comparing sizes and distances. Those measurements are local: they ask what happens over a patch of space. Whether the patch eventually wraps around is a different property, called the topology of space, meaning how its pieces connect. The authors of an early search with the WMAP satellite put it in one line: the local geometry of space constrains, but does not dictate, its topology.

So a flat universe could be infinite, or it could be a three-torus, or one of several other closed flat shapes in which the faces are joined with a twist. A 2022 study lists nine such shapes in which a left hand carried around space comes back still a left hand. Six close up in every direction, two in two directions and one in a single direction.

What we would see if the box were small

In a three-torus, light that leaves a galaxy in one direction can go around the box and reach us from the opposite side. We would see the same galaxy in two or more directions. Galaxies change over billions of years, which makes their copies hard to recognise. A better target is the cosmic microwave background, the faint glow that has travelled to us from the time when the young universe first became transparent. All of that light comes from a huge sphere centred on us, called the last-scattering surface.

If the box is smaller than that sphere, the sphere overlaps its own copy, shifted by one box length. Two spheres that overlap meet in a circle. Every point on that circle sends light to us along two different paths. So the same ring of hot and cold spots appears twice on the sky, in two different places. In 1998 Neil Cornish, David Spergel and Glenn Starkman proposed searching the microwave sky for exactly these matched circles.

The size of the circles carries the size of the box. A box only a little smaller than the visible sphere gives two small circles on opposite sides of the sky. A much smaller box gives large circles and many pairs of them.

What the satellites found

The WMAP team searched for circle pairs placed back to back, on opposite sides of the sky, and nearly back to back. In 2004 they reported no matched circles with a radius larger than 25 degrees. For a wide class of models, that ruled out a universe smaller than 24 billion parsecs across.

The Planck satellite repeated the search with its 2015 maps of temperature and of polarization, the direction in which the light vibrates. Again, no pair of circles stood out above what random patterns produce. For a three-torus built from a cube, Planck concluded at 99 percent confidence that the cube must be at least about 97 percent as wide as the diameter of the visible sphere.

What the answer leaves open

The answer has one real limit. If the box is larger than the visible sphere, the sphere never meets its copy. No circles appear, and the matched-circle test cannot tell a huge finite box from infinite space. A finite flat universe is still allowed. It is just too big to show its repeats in this way.

The Planck limit also applies most directly to the plain cubic box. A 2022 preprint from the COMPACT collaboration points out that in boxes whose faces are joined with a twist, the shortest loop around space can miss our position. Such a loop can be two to at least six times shorter than the diameter of the visible sphere without producing the circles that searches have ruled out. Testing those shapes needs subtler patterns than matched circles.

So a flat universe can be finite. The microwave sky says it is not a small finite box of the simplest kind, and it leaves the larger and the twisted possibilities open.

Sources: Cornish, Spergel and Starkman 1998, circles in the sky, Cornish et al. 2004, WMAP circle search, Planck 2015 results XVIII, background geometry and topology, COMPACT collaboration preprint, limits on orientable Euclidean manifolds.

Drag the box edge and watch repeated images and circle pairs appear.