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 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 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