**Topology of the Universe**

Physicists describe the universe as a
manifold. Mathematicians characterize manifolds in terms of their geometry and
topology. Geometry is a local quantity that measures the intrinsic curvature of
a surface. In a the context of physics, General relativity relates the
distribution of mass in the universe to its geometry and the geometry of the
universe determines the dynamics of the mass. Topology is a global quantity that
characterizes the shape of space (* Measuring the
Topology of the Universe *, Cornish, Spergel, and Starkman). Unlike
the geometry of the universe, the topology of the universe is not constrained by
General Relativity. To understand topology, it is necessary to understand
the possible geometries and curvatures of the universe first.

**Curvature and Geometry**

Under the Standard Big Bang theory, the
universe could have one of three different curvatures: positive, negative
and flat. These different curvatures create universes that are
fundamentally different in their geometry and their destiny. A positively
curved universe can be imagined as a sphere when trying to understand how
geometry behaves in this universe. This statement is somewhat cryptic but
in a positively curved universe, lines donít behave as they do in Euclidean
space. A negatively curved universe is best described as being
hyperbolic. In a flat universe, geometry is Euclidean.

Fig.1: In this picture, the closed universe is first, then the flat,then the open. | Fig. 2: In this picture, the flat universe is first, then the closed, then open. |

In the picture, we see how curvature affects the geometry of space. In positively and negatively curved universes, angles in triangles donít add up to 180 degrees and parallel lines are not really parallel.

The curvature of the universe is a function of the amount of ìstuffî in the universe. By stuff I mean matter both visible and dark and energy. Positive curvature has a matter density (W) of higher than 1. In a negatively curved universe, W<1. In a flat universe, W=1 which is called the critical density.

**Implications of the Different Curvatures**

First, it is important to
understand the implications of what the differences are between these three
geometries. The plot below shows the distance between galaxies as a
function of time.

**Closed Universe**

In a closed universe, which corresponds to a
negative curvature, we would eventually experience something called the ìbig
crunch.î Essentially, this would be the Big Bang in reverse. The
idea is that the universe exploded into existence at time zero and expanded very
rapidly. However, it slowed down because of the gravitational pull of
matter. Eventually, this pull would cause the universe to contract to the
point of a singularity.

**Open Universe**

In an open universe, which corresponds to a positive
curvature, the exact opposite happens and we would experience what is called
heat death. In this scenario, the galaxies fly apart fast enough so as to
escape each otherís gravity. They would eventually fly so far apart so
fast that the galaxies would begin to dissolve along with the stars, planets,
and all other clumped matter. In this case, the universe achieves perfect
homogeneity.

**Flat Universe**

A flat universe, which corresponds to zero or flat
curvature, is a perfect balance between the two others. In this case the
universe expands just fast enough not to be pulled back into a singularity but
just slow enough so as not to suffer heat death. The result is an ever
decreasing, but never stopping, expansion rate of the universe.

Fig.3 |

**Back to Topology**

In the standard Big Bang model, the idea is that
only the matter in the universe affects the geometry of the universe.
However, in inflationary theory, dark energy is added in and it predicts that
the general curvature of space is flat. Inflation is a fairly well
established theory and for the purposes of this website, I will assume that the
prediction that the universe is flat is a reality. A more in depth discussion
of Inflation ( Jennifer Davis' Website) . This assumption is
consistent with the recent finding of WMAP. Because of the change in the
equation for the critical density, knowing the geometry of the universe doesnít
necessarily tell us whether it is open, closed, or at the critical
density. And, if inflation is an accurate depiction of the universe, than
space could be much larger than what we previously thought. Currently we
see the universe as being 3 gigaparsecs across. It is possible that, it is
much bigger and just unobservable.

As said above, WMAP has more or less confirmed that space is flat. It
also has brought up some new data that is both exciting and controversial with
regard to topology. In previous theories the universe is infinite.
In principle, in an infinite universe, the waves from the CMB should appear
randomly around the sky at all sizes. But, according to the new map, there seems
to be a limit to the size of the waves, with none extending more than 60 degrees
across the sky. If one imagines the universe to be a giant guitar string,
then, in this case, it is missing its lowest pitches. This is possibly
because it is not large enough to maintain those wavelengths.

The simplest and therefore the most studied of these possible
topologies is something called a 3-torus. This is an object that is
essentially impossible to visualize but an analogy is a taking a square and
gluing the opposite sides together to form a doughnut. This demonstrated
in the picture to the left. The equivalent to this in 3 dimensions is
doing this same process but with a cube.

The actual shape of this
universe is actually impossible to represent or imagine but we can use an
analogy of a torus. If you exist on

Fig. 5 |

Fig. 4

This type of shape is not the only possiblity there for the
universe. In fact, there are many possiblities and William Thurston and
Jeffery Weeks have been key figures in the development of a catalogue of
possible manifolds (topological shapes) of the universe. I talked about
manifolds above a little bit when describing how physicistsand mathematicians
talk about and represent the shape of the universe. These manifolds are
very difficult to illustrate but are decribed mathematically.

**Implications and Testing**

The consequences of this are quite
profound. If such a topology described the universe, what astronomers might
think is a distant galaxy could actually be the Milky Way -- seen at a much
younger age because the light has taken billions of years to travel around the
universe. The reason for this is because these types of shapes are
multiply connected. This means that if one were to start going in one
direction and keep that direction, they would end up in the spot they
started. Light would do the same thing. If one of these manifolds
described the topology of our universe, the light from distant galaxies would
have intersected many times over and we would find ourselves in a hall of
mirrors of sorts.

Fig. 6 |

An indirect method of testing could be found in the Cosmic
Background Microwave Radiation . At its earliest moments, the universe is
thought to have been nearly uniform, scrunched plasma of photons and various
particles, such as protons, electrons, and neutrinos. About 300,000 years after
the Big Bang, that opaque plasma cooled enough to allow neutral atoms to form.
Instead of being constantly scattered by particles, photons were then free to
cruise the universe essentially unimpeded.

Fig. 8 |

Fig. 7 |

If the universe has a finite size, this expanding sphere "can wrap all the way around and intersect itself," Weeks says.

The intersection of one sphere with itself is seen as a circle. In a finite,
multiply connected universe, an observer at the center of such a sphere would
see the same circle of points in two different directions.

Fig. 9 |

The idea of the fundamental domain is basically the spatial extent of the
finite topology that is our universe. What this all means is that the
sphere of CMB keeps getting bigger and bigger. After a certain point, it
reachs the spatial limit and begins fold back on itself because of the 3-torus
shape of the universe. Because of this, we would see circles in the CMB
like in the picture to the left.

**References** *Measuring the
Topology of the Universe *, Neil J. Cornish, David N. Spergel, &
Glenn D. Starkman *Circles in the Sky,
Detecting the shape of the universe , *Ivars Peterson, Science News
Online, Febuary 21, 1998 *Universe
as Doughnut: New Data, New Debate* , Dennis Overbye, New York Times,
March 11, 2003. *Is
Space Finite?* , Jean-Pierre Luminet , Glenn D. Starkman and Jeffrey R.
Weeks, Scientific American, April 1999 issue

http://www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.htm

**Topology Websites**

1. http://www.nidsci.org/articles/universe_topology.html

2. http://www.astro.princeton.edu/~dns/nas_neg/nas_neg.html

3. http://www.spacedaily.com/news/cosmology-01f.html

4. http://www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.htm

5. http://zebu.uoregon.edu/~imamura/209/apr12/topology.html

6. http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/blackhl.htm

7. http://www.lassp.cornell.edu/GraduateAdmissions/greene/greene.html

8. http://pup.princeton.edu/titles/7324.html

9. http://astro.uchicago.edu/C:/Documents%20and%20Settings/nbbower/Desktop/Topology%20website.html

10.
http://www.sciencenews.org/sn_arc98/2_21_98/bob1.htm

11. http://www.nytimes.com/2003/03/11/science/space/11COSM.html?pagewanted=1

12. http://www.sciam.com/article.cfm?articleID=00065A99-90A6-1CD6-B4A8809EC588EEDF&pageNumber=1&catID=2