Transcript PowerPoint Presentation - The Shape of the Universe

The Shape of the Universe
Topology, Geometry, and
Curvature
The “Big BANG”
• About 14 billion years ago, the Universe
was very hot and very dense
• Since then, it has been expanding and
cooling (the Big BANG does not mean an
explosion)
• After about 4 hundred thousand years, the
Universe cooled enough for photons (light)
to be released
• The Universe was lit up!
The Cosmic Microwave
Background (CMB)
• Discovered by radio-astronomers, Wilson
and Penzias
• Relic of the Big Bang
• These microwaves have been traveling
through the Universe since light was first
released almost 14 billion years ago
• 10% of TV “snow” is CMB
CMB showing temperature variations
CMB Today
• Cooled to almost uniform 2.7 Kelvin (about -455
degrees Fahrenheit)
• Interest in CMB lies in its variations in
temperature of one ten-thousandth of a degree
• Indicates differences in density of early Universe
• Could hold the key to how galaxies were formed
• Denser regions indicate cooler temperatures
where galaxies eventually formed
Last Scattering Surface (LSS)
• Theoretical spherical surface centered at
the Earth where the CMB originates
• Before light was released, photons were
scattered by the free ions, electrons and
protons, hence the name LSS
• The radius of the LSS is estimated at 47
billion light years
Topology of the Universe
• Discovery of the CMB stimulated interest
in the shape of the Universe
• Is it flat, spherical, donut shaped, saddled
shaped?
• And, is it finite or infinite?
• If it’s finite, how big?
• If it’s infinite, well. . . ----
Topology
• Topology is the study of shapes of objects
or spaces
• Two spaces are equivalent if one can be
deformed into the other without breaking
or tearing
• Distance is not important to topology
• Donut = Coffee Cup
Donut (Solid Torus) & Cup
CMB and Topology
• Imagine a finite 2-dimensional Universe
• It could be spherical, or shaped like the
surface of a donut (torus), or the surface of
a donut with several holes
2 Dimensional Sphere (2-sphere)
and
2 Dimensional Torus (2-torus)
Mathematician’s and Cosmologist’s Flat
Torus: glue together or identify opposite
sides of the rectangle
A 2-torus cannot be constructed
in the Euclidean plane
If you were a 2-dimensional person in
a 2-dimensional Universe, you could
not see a complete torus!
Torus Games
developed by Jeff Weeks, Freelance Mathematician
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Open Internet Explorer
Go to www.geometrygames.org
Download “Torus Games”
Open zipped file and click icon
Extract files
Click tic tac toe icon
Click on the window that opens
Play pool on a flat torus
2-Torus in Euclidean 3-Space
• Introduces curvature
Measuring Angles on a Torus
• Draw two large triangles on the tube provided,
one half way around, the other halfway around
on the other side
• Bring the ends of the tubes together so that one
triangle is on the outside and the other on the
inside
• Secure the ends with duct tape
• You now have a solid torus!
• Measure the sum of the angles in each triangle
Curvature of Space in Two
Dimensions
• Positive: sum of angles in triangle is
greater than 180 degrees (outside of a
torus, sphere)
• Zero: sum of angles in triangle equals 180
degrees (flat torus, Euclidean plane)
• Negative: sum of angles in triangle is less
than 180 degrees (inside of a torus,
saddle, or hyperbolic space)
Sphere: Positive Curvature
Hyperbolic Plane: Negative
Curvature
the way that mathematicians like to
view the 2-torus in the plane: Tiling
Building a 3-Torus
• Start with a cube
• Glue together opposite faces of the cube,
analogous to the construction of a 2-torus
CMB and the Shape of the
Universe: Circles in the Sky
• If the Universe is finite and shaped like a torus, then we
might be able to see the last scattering surface on the
faces of the cube
• Think of the LSS as a balloon on the inside of the cube
• As the balloon expands it will press against the faces of
the cube
• We could see this as matched circles on opposite faces
of the cube
• Look to the east, see a circle; look to the west, see the
same circle
• Circles match by pointwise temperature readings
Last Scattering Surface in a Box
A 3-torus cannot be built in Euclidean
3-space, so we can view it by tiling
Flying a Spaceship through a Tiled
3-Torus Universe
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Go to www.geometrygames.org (credit to Jeff Weeks)
Download Curved Spaces
Open zipped file
Click icon
Extract all files
Open folder, click icon
Choose 3-Torus
Click on the window, fly your spaceship!
You can change orientation or direction by holding down
either control or shift or both
• Open Internet Explorer
• Tear down or build up walls using left or right arrows
Is the Universe Shaped Like a
Soccer Ball?
• A theory, championed by French
astrophysicist, J.P. Luminet
• Build the Poincaré Dodecahedral Space
• Glue together opposite faces (pentagons)
of a dodecahedron (soccer ball)
You can also fly your spaceship through a tiled Poincaré
Dodecahedral Space using the “Curved Space” folder on
the website, www.geometrygames.org