StarChild Question of the Month for July 2001

Question:

What is the shape of the universe?

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Answer:

One of the most profound insights of General Relativity was the conclusion that mass caused space to curve, and objects travelling in that curved space have their paths deflected, exactly as if a force had acted on them. If space itself is curved, there are three general possibilities for the geometry of the universe. Each of these possibilites is tied to the amount of mass (and thus to the total strength of gravitation) in the universe, and each implies a different past and future for the universe.

First, let's look at shapes and curvatures for a two-dimensional surface. Mathematicians distinguish 3 qualitatively different classes of curvature, as illustrated in the following image:

image showing a 
sphere which represents a positive curvature surface; a flat piece of 
paper representing a zero curvature surface; and a saddle shape 
representing a negative curvature surface

The flat surface at the left is said to have zero curvature, the spherical surface is said to have positive curvature, and the saddle-shaped surface is said to have negative curvature.

The preceding is not too difficult to visualize, but General Relativity asserts that space itself (not just an object in space) can be curved, and furthermore, the space of General Relativity has 3 space-like dimensions and one time dimension, not just two as in our example above. This IS difficult to visualize! Nevertheless, it can be described mathematically by the same methods that mathematicians use to describe the 2-dimensional surfaces. So what do the three types of curvature - zero, positive, and negative -mean to the universe?

The geometry of the universe is often expressed in terms of the "density parameter", which is defined as the ratio of the actual density of the universe to the critical density that would be required to cause the expansion to stop. Thus, if the universe is flat (contains just the amount of mass to close it) the density parameter is exactly 1, if the universe is open with negative curvature the density parameter lies between 0 and 1, and if the universe is closed with positive curvature the density parameter is greater than 1.

The density parameter determined from various methods such as calculating the number of baryons created in the big bang, counting stars in galaxies, and observing the dynamics of galaxies both near and far. With some rather large uncertainties, all methods point to the universe being open (i.e. the density parameter is less than one). But we need to remember that it is unlikely that we have detected all of the matter in the universe yet.

The current theoretical belief (because it is predicted by the theory of cosmic inflation) is that the universe is flat, with exactly the amount of mass required to stop the expansion (the corresponding average critical density that would just stop the is called the closure density). Recent observations (such as the BOOMERANG and MAXIMA cosmic microwave background radiation results, and various supernova observations) imply that the expansion of the universe is accelerating. If so, this strongly suggests that the universe is geometrically "flat".

In reality, determining the value of the density parameter and thus the ultimate fate of the universe remains one of the major unsolved problems in modern cosmology. The recently (June 30, 2001) launched MAP mission will be able to measure the value definitively within the next 5 years.

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