By: Tony Strazzara
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Why might our universe be finite?
 medieval philosophers gave the first logical
arguments supporting a finite universe
 during the early 20th century, Einstein proposed a
closed, static universe shaped like a hypersphere
 in 1929, Edwin Hubble discovers constant rate of
expansion of universe
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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
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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
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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
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COBE – Cosmic Background Explorer
 launched November 1989
 DMR (differential microwave radiometer) used to
map cosmic radiation
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WMAP – Wilkinson Microwave Anisotropy
Probe
 launched June 2001
 mission: to determine the content, evolution and
geometry of the universe
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Colors
Contrast in clarity
Shape of data
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Temperature
fluctuations
 Large angular scales
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Small and infinite
universe
 ~ 70 billion light years
across
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Let’s change gears…
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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
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Cosmologists consider only 3 types of
geometries for our universe: hyperbolic
(negative curvature), elliptic (positive
curvature), and Euclidean (zero curvature)
The Hyperbolic Plane
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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…
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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
T3 space
T2 space
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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”?
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If observed space (LSS) is larger than physical
space, expect correlations in CMB
 Balloon-cube analogy
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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
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Think about it first in a 2-manifold…
 Torus
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Now for a 3-manifold…
 3-Torus
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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?
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By constructing loops in the fundamental group, we
can predict the topology of the universe
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Distortion due to a negatively curved universe
 The lines that light follow in hyperbolic space
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Circles in the sky will be distorted into ovals because
of this curvature of space
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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
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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
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Would mean universe has slightly positive curvature
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3-Torus
Mirrored Dodecahedron
Poincare Dodecahedron
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7-year map released earlier this year
Neutrinos
Zero curvature
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Cornish, Neil J., Spergel, David N., and Starkman, Glenn D. “Measuring the Topology of the
Universe.” <http://www.pnas.org/content/95/1/82.full.pdf+html>.
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?”
<http://arxiv.org/PS_cache/astro-ph/pdf/9510/9510109v2.pdf>.
Levin, Janna J., Barrow, John D., Bunn, Emory F., and Silk, Joseph. “Flat Spots: Topological
Signatures of an Open Universe in COBE Sky Maps.” <http://arxiv.org/PS_cache/astroph/pdf/9702/9702242v1.pdf >.
Levin, Janna. “Missing Lorenz-boosted Circles in the Sky.” <http://arxiv.org/PS_cache/astroph/pdf/0403/0403036v1.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.” <http://arxiv.org/PS_cache/astro-ph/pdf/0402/0402608v4.pdf>.
Greason, Michael R. “Cosmic Background Explorer.”
<http://lambda.gsfc.nasa.gov/product/cobe/>.
Greason, Michael R. “Wilkinson Microwave Anisotropy Probe.”
<http://lambda.gsfc.nasa.gov/product/map/current/>.
Muir, Hazel. “Tantalising Evidence Hints Universe is Finite.”
<http://www.newscientist.com/article/dn4250-tantalising-evidence-hints-universe-isfinite.html>.