The question of the Universe’s shape has been one of the most important in modern cosmology. For many years, there has been a great discussion among astronomers regarding this problem. Today, according to the most up to date measurements, it is widely accepted that the shape of the Universe is flat.

According to General relativity, the local geometry (the curvature of the observable universe at any arbitrary point) of the Universe is described by what we call the Friedmann-Lemaître-Robertson-Walker (FLRW) model. This model, described by the Friedmann equations provides a curvature (‘geometry’) of the universe. Here, the matter within the Universe is modeled as a perfect fluid. Mass causes space to curve, this is one of the most important conclusions of General Relativity. The shape of the universe is then determined by the fight between the pull of gravity and the momentum of expansion.

Considering that most of the matter we know has a low pressure, the fate of the universe is governed by its density. In the FLRW model, the density parameter Ω is related to the curvature of space. This parameter is defined as the ratio of the actual density of the universe and the critical density (the required density for the universe to be flat).

This provides three possible scenarios:

- If Ω = 1, the curvature of the universe is zero, and the universe is flat.
- If Ω > 1, the curvature is positive, and the universe is closed.
- If Ω < 1, the curvature is negative, and the universe is open.

In the first case, there is exactly enough mass to stop the expansion, after an infinite amount of time. In other words, the universe will expand forever and has no boundaries.

In the second case, there is enough mass to slow down and stop the present expansion. The universe is not infinite, but has no boundaries (you can picture it as a sphere, you cannot call any point on the surface of the sphere an end). Eventually, the universe will collapse producing what we call Big Crunch.

In the third case, there is not enough mass to stop the expansion. The universe will expand forever and has no boundaries.

One of the most powerful ways to assess the curvature of the universe is to study the cosmic microwave background radiation (CMB – the remnant energy left over from the formation of the universe, according to the Big Bang theory). In 2001, the Boomerang experiment measured the CMB radiation of a part of the sky and showed that the universe was flat within 15% accuracy. This results were confirmed in 2008 by the WMAP experiment, which also showed the universe to be flat, within 2% accuracy.

Recently, the satellite Planck has been launched to perform new measurements of the CMB, and should provide even more accurate results.

While these observations indicate that the universe seems to be flat, others have shown that it is expanding at an accelerated rate (which also favors a geometrically flat universe), implying the existence of a form of matter with a strong negative pressure, called dark energy.