# Harmonics Echoing Across Time and Space: A Summary of Research on the Topology of the Universe Open Access

## Lykken, Sara Loraine (2010)

Permanent URL: https://etd.library.emory.edu/concern/etds/5138jf01c?locale=en
Published

## Abstract

Harmonics Echoing Across Time and Space:
A Summary of Research on the Topology of the Universe
By Sara Lykken

On June 30, 2001 NASA launched its Wilkinson Microwave Ansiotropy
Probe (WMAP) to study the Cosmic Microwave Background (CMB) radi-
ation, radiation from the plasma that filled the universe until just 380,000
years after the Big Bang. The information collected by the WMAP satellite
provides clues as to the universe's topology. Specifically, early WMAP data
suggested the near disappearance of temperature fluctuations in the CMB ra-
diation on large angular scales, going completely against what was predicted
by an infinite Euclidean model for the universe. Researchers attempted to
explain the missing large-scale fluctuations by suggesting the possibility of a
small, finite-volume universe. Observations about the local geometry of our
universe suggest a spherical, Euclidean or hyperbolic universe and, primar-
ily for reasons of practicality, research has focused on spherical 3-manifolds.
Comparing observed fluctuations in the CMB radiation to those predicted
by various spherical spaces, some advocate the Poincare dodecahedral space
as a model for our universe. Recently, however, new data has been released
that may weaken the case for the dodecahedral space. This paper seeks to
explain and expand upon these topological investigations with the intention
of making them accessible to a wider audience, in order to facilitate dialogue
with specialists in other relevant areas.

Contents

1 Introduction..........................................1
2 The geometry of our universe..................3
3 Constructing a model of the universe.......11
4 Hearing the shape of the universe...........20
5 A spherical universe?.............................27
6 Searching for a suppressed quadrupole....33
7 Do physical observations support the predictions of a dodecahedral
model?...................................................37
8 New evidence.......................................40
9 Appendix A..........................................43
10 Appendix B........................................45

List of Figures

1 An octagon in the hyperbolic plane . . . . . . . . . . . . . . . . . . . 6
2 Straight lines in the hyperbolic disk model . . . . . . . . . . . . . . 7
3 Forming an octagon with 45 degree angles . . . . . . . . . . . . . .7
4 Tiling the hyperbolic plane with octagons . . . . . . . . . . . . . . . 8
5 A piece of the hyperbolic plane . . . . . . . . . . . . . . . . . . . . . ..8
6 A Flattened piece of the hyperbolic plane . . . . . . . . . . . . . . . 9
7 Triangles in E2, S2 and H2 . . . . . . . . . . . . . . . . . . . . . . . . 10
8 A circle on a sphere's surface . . . . . . . . . . . . . . . . . . . . . . 17
9 The CMB power spectrum . . . . . . . . . . . . . . . . . . . . . . . . ..22
10 Constructing a lens space . . . . . . . . . . . . . . . . . . . . . . . . 35
11 Seeing large-scale Fluctuations in a nite-volume universe . . 44

List of Tables

1 Possibilities for mth-degree monomials on the 3-sphere . . . . .24
2 Finite Symmetry Groups of S2 . . . . . . . . . . . . . . . . . . . . . . 29