The Shape of the Universe

1. The Shape of the Universe Topology, Geometry, and Curvature

2. 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!

3. 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

4. CMB showing temperature variations

5. 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

6. 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

7. 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. . . ----

8. 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

9. Donut (Solid Torus) & Cup

10. 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

11. 2 Dimensional Sphere (2-sphere)and 2 Dimensional Torus (2-torus)

12. Mathematician’s and Cosmologist’s Flat Torus: glue together or identify opposite sides of the rectangle

13. 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!

14. Torus Gamesdeveloped by Jeff Weeks, Freelance Mathematician • 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

15. 2-Torus in Euclidean 3-Space • Introduces curvature

16. 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

17. 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)

18. Sphere: Positive Curvature

19. Hyperbolic Plane: Negative Curvature

20. the way that mathematicians like to view the 2-torus in the plane: Tiling

21. Building a 3-Torus • Start with a cube • Glue together opposite faces of the cube, analogous to the construction of a 2-torus

22. 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

23. Last Scattering Surface in a Box

24. A 3-torus cannot be built in Euclidean 3-space, so we can view it by tiling

25. Flying a Spaceship through a Tiled 3-Torus Universe • 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

26. 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)

27. You can also fly your spaceship through a tiled Poincaré Dodecahedral Space using the “Curved Space” folder on the website, www.geometrygames.org