8: Cosmic Topology
Cosmic topology can be described as the effort to determine the shape of our universe through observational techniques. In this chapter we discuss two programs of research in cosmic topology: the cosmic crystallography method and the circles-in-the-sky method. Both programs search for topology by assuming the universe is finite in volume without boundary. The chapter begins with a discussion of three-dimensional geometry and some \(3\)-manifolds that have been given consideration as models for the shape of our universe.
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- 8.1: Three-Dimensional Geometry and 3-Manifolds
- Euclidean geometry is the geometry of our experience in three dimensions. Planes look like infinite tabletops, lines in space are Euclidean straight lines. Any planar slice of 3-space inherits two-dimensional Euclidean geometry. The Poincaré disk model of hyperbolic geometry may also be extended to three dimensions. Three-dimensional elliptic geometry is derived from the fact that the 3-sphere consists of all points in 4-dimensional space one unit from the origin.
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- 8.2: Cosmic Crystallography
- Imagine once again that we are two-dimensional beings living in a two-dimensional universe. In fact, suppose we are living in the torus in Figure 8.2.1 at point E (for Earth). Our world is homogeneous and isotropic, and adheres to Euclidean geometry. Our lines of sight follow Euclidean lines. If we can see far enough, we ought to be able to see an object, say G (for galaxy), in different directions. Three different lines of sight are given in the figure.
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- 8.3: Circles in the Sky
- Immediately after the big bang, the universe was so hot that the usual constituents of matter could not form. Photons could not move freely in space, as they were constantly bumping into free electrons. Eventually, about 350,000 years after the big bang, the universe had expanded and cooled to the point that light could travel unimpeded. This free radiation is called the cosmic microwave background (CMB) radiation, and much of it is still travelling today.
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- 8.4: Our Universe
- Our universe appears to be homogeneous and isotropic. The presence of cosmic microwave background radiation is evidence of this: it is coming to us from every direction with more or less constant temperature. The assumptions of isotropy and homogeneity are remarkably fruitful when one approaches the geometry and topology of the universe from a mathematical point of view. The mathematical point of view gives us our candidate geometries.