Graduate Program

Natural Sciences

Degree Name

Master of Science (MS)

Semester of Degree Completion

2000

Thesis Director

Keith Andrew

Abstract

The cosmic microwave background explorer, COBE, and the Balloon Observations of Millimetric Extragalactic Radiation and Geophysics, BOOMERanG, have collected data from the universe and detected relic anisotropies indicative of one of the earliest events of the universe, decoupling. Hidden in the correlation between these temperature fluctuations is the signature of the global shape, or topology, of the early universe. It is possible to calculate the temperature fluctuations as due to primeval adiabatic density temperature fluctuations from the Sachs-Wolfe effect, which contains a topological term. Here we investigate some of the spaces and how they affect the microwave background.

A large class of spaces can be understood with some tools of topology associated with the way curves and volumes divide a space. The three tools of interest are the Euler Characteristic, the Betti numbers and fundamental domains. We intend to demonstrate the relationship between topology and anisotropy by creating a general equation that contains a topological term based on a topological invariant, the Betti Numbers of a manifold, and a term based on the temperature fluctuations.

We will also extend the work of Silk on classifying the finite flat spaces through the use of a new three-dimensional plotting technique. These graphs demonstrate the inadequacy of the current CMB data's topological predictive properties. As many shapes can produce the observed peaks. We will also extend the work of Inoue on classifying compact hyperbolic manifolds in two ways. First we demonstrate that the compact hyperbolic eigenmodes can be represented by a pFq function, and do not need to be calculated individually. Secondly, we extend the power spectrum plot the high angular resolution, large l, and compare compact hyperbolic manifolds to the observed peak in the BOOMERanG data at l = 200.

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