Geometry of the Universe

After Einstein invented his general theory of relativity, Friedman dveloped the first models for the structure of universes based on GR (Lemaitre also investigated various models for universes). Based on GR, Friedman found three basic styles of universe. We demonstrate some properties of these universes with two-dimensional analogies. Recall that our Universe currently has four dimensions (which is real hard to visualize). We have flat space (Euclidean), postive curvature space (e.g., a sphere), and negative curvature space (e.g., a saddle) universes. Graphically, we have:

• positive curvature and negative curvature
• properties 1
• properties 3

Abstract as the above concepts are, our Universe has one of these shapes (topologies), but so what. Well, the different shapes are indicative of the ultimate fate of the Universe; the ultimate fate is different for each style of universe. This means that if we can determine the geometry (shape) of the Universe, we can infer what is going to the Universe in the distant future.

We now show what each solution looks like in terms of what is referred to as the scale factor for the Universe, R(t). The scale factor, R(t), tells us how much bigger the Universe is today than it was yesterday and so on.

• Scale Factor ===> D = R(t) x D(in the past) where D(in the past) represents some constant number.

In terms of the evolution of the scale factor, R(t), the various solutions look like:

The positive curvature universes (spheres) correspond to closed universes, negative curvature universes (saddles) correspond to open universes and flat universes correspond to the critical universe

In terms of a Hubble plot, we have that:

A great deal of effort is now directed toward determining the shape of the Universe. We will spend a fair amount of time on one method (later), but for now let me touch on a method based on things we have already discussed -- namely looking at geometric things.

Shape of the Universe

• properties 2

  Depending upon the geometry of the Universe, the number density of galaxies at large redshift (===>large recession velocity ===>large distance) should depend upon the geometry of the Universe. Why? Well, just as the area of a circle depends upon the geometry of the Universe, the volumes of objects may also depend upon the geometry of the Universe. The difference in the way volumes depend upon distance will affect the way densities depend upon distance. Tests for this effect have been carried out. The results are interesting, but not conclusive.

• properties 1

  Depending upon the geometry of the Universe, the paths of parallel lines can cross or diverge, A consequence of this is that if measures the angular sizes of distant galaxies, they do necessarily have to decrease as 1/distance (as they would in a flat universe). This effect has also also been studied. The results are again suggestive but not conclusive.