Topology of the Universe
By Nicholas Bower

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.

So if the universe is not infinite, then what does it look like?
The simplest topology to understand is the Euclidean geometry with infinite space and most maintain that this is also the most natural and hence the most probable.  However, some cosmologists have a hard time calculating how an infinite universe could have appeared in that kind of space.  ìNature, they contend, might have had an easier time making a small "compact" universe than an infinite one, and they assume Nature would take the easy way outî (Dennis Overbye, New York Times, 3/11/03). 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
the surface of torus, it seems flat locally just like the surface of the Earth.  If one were to walk in any direction for long enough they would end up back where they began.  This concept can be likened to the spacewar game in two dimensions.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  In that game, if a spaceship were to leave one side of the space it would reenter on the other side of the screen.

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
The idea that astronomers would actually be able to see all the way around the universe and find similarities between images of galaxies or quasars that appear in different directions, representing different times in their past, has proved extremely difficult.  This is mainly because the light takes so long to get all the way around the universe; the galaxies have evolved too much to be recognizable.

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
As viewed from Earth, which is immersed in this cosmological radiation, the photons appear to come from a sphere centered on the observer. Called the surface of last scattering, that sphere has a radius equal to the speed of light multiplied by the travel time since those photons' last interaction with matter. Because of the subsequent expansion of the universe, the photons have shifted to lower wavelengths, and they are now observable as the cosmic microwave background.
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
Today the CMB appears as a giant sphere of radiation with us at the center of it (we are not in a special place - all locations see the CMB as a sphere of radiation with the observer at the center). If the universe is a 3-torus and if the radius of the CMB sphere is sufficiently large, then it will not fit within the fundamental domain.

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