The Topology of the Universe

Ideas regarding the shape of the universe are probably as old as humanity. At least since ancient Greek times, people have wondered whether space is finite or infinite in extent. Since only Euclidean geometry, describing an unbounded space, was known at the time, it was tacitly assumed that niteness implied the existence of a boundary. Although these matters were of some philosophical interest, the constraints of technological and mathematical possibility would render them inaccessible to physical inquiry for centuries to come. Luckily, around 1830 non-Euclidean geometry was discovered and with the formulation of Riemannian geometry in 1854, the theoretical framework was finally present to describe spaces and their shapes. The course of history seems to prove a striking human preoccupation with at Euclidean space over curved non-euclidean spaces. During the 20th century, the discovery of General Relativity inspired great skepticism toward the Euclidean model, though in the end it seems to have managed to withstand all attempts at its falsification. Recent empirical methods, especially the WMAP, have ruled out more and more possible curvature, but Euclidean space remains safely inside of our confidence intervals. The evidence in favor of at space is at best as compelling as the evidence against it, however, and the only thing we can be certain of is that our space is very close to being at (compared to what?). In 2003, an analysis of periodicity in the cosmic microwave background (CMB) led Jean-Pierre Luminet to suggest that the universe is a topological space known as the Poincaré dodecahedral space (PDS). As will be shown, this implies a spherical geometry, in stark contrast to the concordance model of cosmology. In this thesis, I will explore the credibility of the Poincaré dodecahedral space as a model for our universe and I will also attempt to illuminate the methods by which such a model can be veried, or more realistically falsied. To accomplish these goals, section 2 will be a quick introduction to 3-dimensional topology and its geometries. For the sake of brevity, this thesis will then focus on the Poincaré dodecahedral space, disregarding all other geometries and manifolds. Section 3 is dedicated to proving that the Poincare dodecahedral space does in fact result from identifying faces of a dodecahedron in a certain way. To do this, I will systematically rule out all other possibilities for the shape of the tessellation of S3 representing the PDS. In section 4, I will try to explain the merits of the PDS model in explaining physical data, meaning mostly the WMAP data on the CMB. During this section, four known ways of detecting cosmic topology will be addressed; searching for duplicates, circles in the sky, cosmic crystallography, and the CMB quadrupole. This This section will be the climax to my argument and will hopefully put the previous sections to good use. The main results and lack thereof will, as per convention, be stated in section 5, which will conclude the paper.