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What is the Shape of the Cosmos

Geometry of Curved Space

The large scale geometry of the universe is governed by Einstein's General Theory of Relativity. Einstein showed that gravity curves three-dimensional space, and that space in turn moves matter. For the universe as a whole, the shape of the curvature depends on the average density of the matter.

If the average density of matter in the universe is greater than the critical density, the force of gravity will eventually rein in expansion and cause the universe to collapse upon itself. In this case, the universe is said to be positively curved, and Omega, the ratio of the average density to the critical density, is greater than 1.

Conversely, if the average density of matter in the universe is less than the critical density, gravity will lose its grip on matter and the universe will expand forever. This negatively curved universe is defined by an Omega less than one.

If Omega is exactly one--that is, if the average density of the universe is equal to the critical density--then the universe will expand to a maximum density and remain there for eternity. This universe is flat; it has zero curvature.

Cosmos Curvatures

Now three dimensional curved space is difficult to visualize, but we can illustrate the curvature in two dimensions. A positively curved universe is like the surface of a sphere; a negatively curved universe, like a saddle. A universe with zero curvature is, not surprisingly, like a plane.
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If you could draw a triangle by reaching far enough into space to draw lines connecting three far-flung galaxies, you could determine the curvature of the universe. The angles of a triangle in a negatively curved universe would add to greater than 180 degrees; those of a triangle in a positively curved universe, to less than 180 degrees. In a flat universe, familiar Euclidean geometry applies, and the angles of the triangle add up to exactly 180 degrees.

Can We Measure the Curvature?

In principle, cosmologists can determine the curvature of three dimensional space by similar means using volumes rather than areas. Unfortunately, geometric methods have proven impractical because they rely on a uniform distribution of galaxies throughout the universe which does not evolve with time--assumptions which are false since observations indicate that the galaxies have a finite age and have changed over the eons.

Back to Omega

Consequently, cosmologists attempt to infer the curvature by measuring Omega--not a trivial task, as it turns out. All of the luminous matter in the universe--everything that can be detected through telescopes--adds up to only one half of one percent of the critical density, or Omega = 0.005. But there is a lot of dark matter--perhaps in the form of black holes, dwarf stars, or exotic particles--in the universe. How do we know it's there?

Just like the wind, we can't see dark matter, but we can see its effects. By exerting its gravitational influence, dark matter affects the motions of stars and gases in nearby galaxies, and on the motions of galaxies and galaxy clusters themselves. Cosmologists analyzing these motions have come up with Omega values of 0.1 to 1.

Inflation theory predicts Omega to be exactly 1, implying that the universe is flat, or has zero curvature. Whereas a positively curved universe is finite (like the surface of a sphere), both negatively curved and flat universes extend forever. Therefore, if Omega really is 1 throughout the cosmos, then the universe we inhabit is infinitely large, which means we can see only an infinitesimally small fraction. No wonder we've detected so little matter in the universe!

Is the Cosmos Open or Closed?

The shape of the cosmos and its evolution and fate are intimately bound up with each other. What will the universe look like billions of years from now?

Forward to Fate of the Cosmos
Return to The Flatness Problem
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Copyright, (c) 1995: Board of Trustees, University of Illinois

NCSA. Last modified 11/2/95.