By: Tony Strazzara

 Why might our universe be finite?  medieval philosophers gave the first logical arguments supporting a finite universe  during the early 20 th century, Einstein proposed a closed, static universe shaped like a hypersphere  in 1929, Edwin Hubble discovers constant rate of expansion of universe

 Galaxy clusters slowly growing farther apart  If distance between them is increasing today, then they were closer together in the past  Raisin bread analogy  Roughly 13.7 billion years ago

 Discovered accidentally in 1964 by two radio astronomers, the CMB is electromagnetic radiation filling the universe  Photons emitted from hydrogen plasma roughly 300,000 years after the big bang occurred  Thermal spectrum of ~2.725 K

 The surface of last scattering (LSS)  Observing the CMB means looking back in time  Looking every direction at one instant in time produces a sphere

 COBE – Cosmic Background Explorer  launched November 1989  DMR (differential microwave radiometer) used to map cosmic radiation  WMAP – Wilkinson Microwave Anisotropy Probe  launched June 2001  mission: to determine the content, evolution and geometry of the universe

 Colors  Contrast in clarity  Shape of data

 Temperature fluctuations  Large angular scales  Small and infinite universe  ~ 70 billion light years across  Let’s change gears…

 A manifold is a space that on a local scale resembles Euclidean geometry of a specific dimension while on a global scale may be more complicated  What is a 2-manifold?  3-manifolds  Cosmologists consider only 3 types of geometries for our universe: hyperbolic (negative curvature), elliptic (positive curvature), and Euclidean (zero curvature)

The Hyperbolic Plane

 What is a 3-torus?  To an observer, line of sight straight ahead eventually leads to seeing the back of one’s own head  And looking up or the right?  We’ll take a look later…

 No – data of CMB from COBE disproves a cubic 3-torus (T3 space)  In fact, cosmologists have ruled out the possibility of any toroidal model  T1 and T2 spaces T1 space T2 space T3 space

 Due to the nature of most other 3-manifolds, our universe is presumably negatively curved  Cosmologists can propose a topology for the universe, deduce what the CMB should look like, then observe how well COBE/WMAP data matches up  So how does this data “match up”?

 If observed space (LSS) is larger than physical space, expect correlations in CMB  Balloon-cube analogy  Correlation is in form of circle pairs  2 identical circles – one from sphere leaving a face of the manifold and one from sphere entering opposite face back into the manifold  Seen as 2 circles with identical variations in temperature fluctuations

 Think about it first in a 2-manifold…  Torus

 Now for a 3-manifold…  3-Torus

 Distance between points in a circle pair is a path  Note – if you choose the same point in both circles, one as a starting point of the path and the other as the ending point, you have constructed a loop! Why?  By constructing loops in the fundamental group, we can predict the topology of the universe

 Distortion due to a negatively curved universe  The lines that light follow in hyperbolic space  Circles in the sky will be distorted into ovals because of this curvature of space

 PDS is a possible manifold for topology of universe  12 faced polyhedron  Each face is a pentagon  Glue opposite faces to each other with a minimal clockwise turn

 New evidence supporting the possibility  Temperature fluctuations around 12 dodecahedrally spaced circles of radius ~11 degrees found in WMAP correspond unusually well  dodecahedrally - phase shift of 36 degrees  Would mean universe has slightly positive curvature

 3-Torus  Mirrored Dodecahedron  Poincare Dodecahedron

 7-year map released earlier this year  Neutrinos  Zero curvature

 Cornish, Neil J., Spergel, David N., and Starkman, Glenn D. “Measuring the Topology of the Universe.”.  Oliveira-Costa, Angelica de, Smoot, George F., and Starobinsky, Alexei A.. “Can the Lack of Symmetry in the COBE/DMR Maps Constrain the Topology of the Universe?”.  Levin, Janna J., Barrow, John D., Bunn, Emory F., and Silk, Joseph. “Flat Spots: Topological Signatures of an Open Universe in COBE Sky Maps.”. ph/pdf/9702/ v1.pdf  Levin, Janna. “Missing Lorenz-boosted Circles in the Sky.”. ph/pdf/0403/ v1.pdf  Roukema, Boudewijn F., Lew, Bartosz, Cechowska, Magdalena, Marecki, Andrzej, and Bajtlik, Stanislaw. “A Hint of Poincare Dodecahedral Topology in the WMAP First Year Sky Map.”.  Greason, Michael R. “Cosmic Background Explorer.”.  Greason, Michael R. “Wilkinson Microwave Anisotropy Probe.”.  Muir, Hazel. “Tantalising Evidence Hints Universe is Finite.”. finite.html