## Sunday, May 8, 2011

### Manifolds: The Shape of the Universe III

This is the final post of the Manifolds Series and the third concerning The Shape of the Universe (see the first of the entire series, or the first concerning The Shape of the Universe).

It was previously discussed that the constant Ω represents the ratio of the Universe's actual density to the so-called critical density which makes the Universe Euclidean. As of yet, the most accurate observation of the density of the observable Universe yields an Ω of 1.02, with a possible error of just over .02. This suggests that the Universe is most likely to be elliptic. However, none of the geometries can be eliminated yet, and other clues will most likely be required to definitively determine its shape.

At the moment, it seems that a Universe with an edge can be ruled out definitively, as the presence of boundary would disturb the isotropy (similarity of the view in each direction from any point in the Universe) which seems to be necessary for a Universe of constant density to form. Since this is most likely the case, a Universe with an edge can be ignored.

This leaves two possibilities. Either the Universe is infinite (no identical image points) or it is finite (with at least one image point in the night sky) but without boundary. The location and distance of these images would determine the shape of the Universe.

Take, for instance, that the Ω value is exactly 1.02, as predicted by current measurements. This implies an elliptic Universe. If the Universe is a 3-sphere, the leading theory for an elliptic structure, then a value of 1.02 would imply a radius of 98 billion light-years, meaning that, if one was to look 98 billion light-years into space, they would see the same view in every direction, namely the diametrically opposite pole. In other words, an expanding ball of space in the Universe would first intersect itself when it reaches this radius, known as the injectivity radius for the given Universe. These similar images in all directions would be very easy to spot, as they would be of the same distance, and therefore the same age.

However, for this particular value of Ω, these images are beyond our ability to see. The Universe is only about 13.7 billion years old, and we can therefore only see that far. Despite this, the most distant objects are seen as they were billions of years ago, and they actually have moved farther away since then. Extrapolating backward from the current rate of expansion, one finds that our observable Universe actually measures 46 billion light-years in radius. Although this is a significant portion of the previously discussed 3-sphere, it is by no means enough to identify the shape of the Universe through images.

One encounters interesting phenomena if the speed of light is allowed to reach infinity in an idealized Universe, making it so that multiple images could be detected. If this was the case, the spacing and distance of images, and shape of the "shells" of the images would identify the manifold. Consider the example below.

In this particular case, images of the Earth can be seen in all directions, and all of the same age and appearance. This is because the speed of light is supposed to be infinite, and the light from all images instantly reaches Earth. Each exists at the center of a dodecahedral "cell", each of which, on its own, represents the entirety of the Universe. The other cells outside of the center one are images. The manifold in question is known as the Poincare dodecahedral space. This is an elliptic manifold with properties similar to that of the 3-sphere. Its construction is shown in detail below.

The fundamental polyhedron for the Poincare dodecahedral space is a dodecahedron, hence the name. This polyhedron has twelve faces, so each face can be connected to the one opposite from it. However, to pair any face with its opposite requires a rotation of one of the faces, for they are not in the same orientation, despite being the same size. The faces are pentagons, and the opposite ones are misaligned by a 1/10 turn. Therefore, by first rotating the indicated face A counterclockwise by a 1/10 turn so that the A1 edges line up, one can connect the faces. This procedure is repeated for all of the pairs of opposite faces. Note that a 3/10 or 5/10 turn produces a completely different manifold! Therefore, choices of rotation and the choice of which pairs of faces are to be attached are both are crucial to determining a manifold.

With this construction in mind, further insight can be gained into the Poincare dodecahedral Universe shown above. It is now clear why the cells are dodecahedra, and that the distortion of these cells is by nature of the manifold being elliptic, as it does not "fit" into Euclidean three-dimensional space without distortion. Second possibly only to the 3-sphere, the Poincare dodecahedral space has the most following of any theory for the shape of an elliptic Universe.

For other geometries and manifolds, the image-finding method is even more difficult than that of the 3-sphere case. For a hyperbolic manifold of given curvature, the deviation of the Ω value from 1 produces a higher injectivity radius then an elliptic manifold of the same deviation, and Euclidean manifolds have images that are spaced unevenly and are at many different distances. (see also the discussion of the 3-torus Universe, found here)

Finally, observations of how forces, particularly gravity, affect objects may be helpful in determining the Universe's shape. At (relatively) small scales, when comparing stars, galaxies, and even superclusters, the gravitational pull of these massive objects distorts the local geometry of the Universe. However, when one considers the entire observable Universe, gravity's effect assumes a more uniform state.

To begin an analysis on how gravity's effect is determined by the shape of the Universe, is is useful to consider the mass distribution in its very early stages. The best source for this information is in the Cosmic Microwave Background Radiation. This radiation was emitted approximately 380,000 years after the Big Bang, by the matter present at the time, and, even at that stage, there were slight discrepancies in density and therefore temperature that gravity slowly molded into the structures we see today. The above image is of the temperature variances in this plasma, the precursor of all that we know in the Universe.

But exactly how did this process occur? Different universes and the gravitational differences between them have been analyzed in previous posts, but many of these properties were concerned with the effects of gravity traveling all around the Universe, and if it is of sufficient radius, these effects are not visible. Also, the most solid evidence thus far points to a lack of both images and these effects points to a Universe of very little curvature, if any. The local properties of our Universe closely resemble an infinite flat one, with just a hint of positive curvature.

In conclusion, the Universe is most likely to be elliptic, in the form of a 3-sphere, as this is the simplest of 3-manifolds, and it is not known how early Universe phenomena could have contributed to turning the Universe into a more complicated manifold, such as the Poincare dodecahedral space. The density measurements, with image and gravity evidence taken in mind yield an Ω probably between 1.01 and 1.02. The radius of the Universe is therefore very large, possibly over 100 billion light-years, and since this figure is constantly increasing, it is unlikely that the shape of the Universe can ever be determined through the image method alone.

The study of manifolds and topology is a broad and insightful area of mathematics that the above series of posts has only touched upon. The potential of manifolds in projection, mappings, the abstract and elegant constructions, and many other aspects of manifolds makes it an important area of study, which may even reveal what type of Universe we live in.

Sources: http://www.ams.org/notices/200406/fea-weeks.pdf, http://en.wikipedia.org/wiki/Homology_sphere#Poincar.C3.A9_homology_sphere,