Published on *Astronomy 801: Planets, Stars, Galaxies, and the Universe* (https://www.e-education.psu.edu/astro801)

Additional reading from www.astronomynotes.com [1]

Given the following information:

- Hubble's Law implies the Universe is expanding;
- We observe the galaxies in the Universe to lie in a filament / void structure, but overall to be distributed on the largest scales in a homogeneous and isotropic way;
- General Relativity describes the behavior of space-time in the Universe;

Astronomers have adopted a model known as the **Hot Big Bang** model to describe the Universe that incorporates the information above. You can state the model pretty concisely—we propose that the entire visible Universe and all of its contents was contained in a tiny region that was originally the size of a pinpoint. At one instant in time, that very hot, very dense point began expanding, and it is now much larger and much cooler than it was at the beginning.

Using the mathematical formulation of General Relativity, you can show that for a homogeneous, isotropic Universe, the geometry of the Universe must be either flat (that is, it obeys the laws of geometry that we all learned in math class), positively curved, or negatively curved. These possibilities are illustrated (to the best that they can be approximated in a flat illustration) below.

Credit: NASA / WMAP [3]

Recall that in the flat geometry that you learned in school, you were taught that the sum of the interior angles of a triangle equals 180 degrees. In the bottom part of the illustration, you see a flat plane with a normal triangle drawn on the plane. The top part of the illustration represents a closed, positively curved or spherical geometry. In the spherical geometry, the sum of the interior angles of a triangle add up to more than 180 degrees. The middle of the illustration represents an infinite, negatively curved or hyperbolic geometry. In this geometry, the sum of the interior angles of a triangle add up to less than 180 degrees. In principle, if there was a large enough triangular object in the universe and you could measure its interior angles, you could determine the local geometry of the Universe. There are other tests you could also conceive. For example, in our familiar flat geometry, two parallel lines remain parallel for their entire length. In a spherical geometry, parallel lines converge, and in a hyperbolic geometry parallel lines diverge. Note that if the Universe is spherical, if you could travel in one direction long enough, you would return to your original position. In either of the other two cases, you would never return to your original position.

You should keep in mind that these different geometric effects are observable in our universe. If we can detect them directly, we can determine the details of the shape of our universe. One more realistic example is to consider using a standard ruler to measure how distance and angular size are related. Consider an analogy on Earth: if you look out your window and see a nearby tree and a more distant tree, even if they are the same real size, the nearby tree will appear larger to you. We know that in our familiar flat geometry, as an object gets farther away from us, it looks smaller. However, in the positively curved geometry, an object with a given size will look smaller and smaller as it gets progressively farther away, but at some critical distance, it will start to look larger again! So, if you can identify an object with a known size (say a galaxy), and you see that the sizes of galaxies shrink as they get farther away from us, but then their apparent sizes start to grow again for all galaxies past a certain redshift, we must live in a positively curved universe. In practice, this test is very hard to do, because we do not know of a good object that can be seen at large distances and is always the same size (that is, the diameter of galaxies is not constant enough to do this test well).

Another way to determine the geometry of the Universe is to try to measure the total quantity of matter it contains, since the amount of matter determines the geometry of the Universe. If we designate a critical density and we call that value ${\rho}_{\text{crit}}$ :

- If the measured density of the universe, ${\rho}_{\text{ave}}>{\rho}_{\text{crit}}$ then the geometry of the Universe will be spherical, and the Universe will be closed.
- If the measured density of the universe, ${\rho}_{\text{ave}}<{\rho}_{\text{crit}}$ then the geometry of the Universe will be hyperbolic, and the Universe will be open and infinite.
- If the measured density of the universe, ${\rho}_{\text{ave}}={\rho}_{\text{crit}}$ then the geometry of the Universe will be flat, and the Universe will be infinite.

In the illustration above, the geometries are labeled with, for example, ${\Omega}_{0}=1$ . In this notation, ${\Omega}_{0}=\text{}{\rho}_{\text{ave}}/\text{}{\rho}_{\text{crit}}$ , so if ${\Omega}_{0}=1$ , this means the same thing as ${\rho}_{\text{ave}}=\text{}{\rho}_{\text{crit}}$ .

Finally, we should address some limitations of the Big Bang model and some misconceptions to watch out for when discussing the Big Bang model. These are:

- Because the Big Bang occurred approximately 13.7 billion years ago, we are only able to observe the Universe within about 13.7 billion light years of Earth. This boundary is referred to as our
**horizon**. The Big Bang model only describes the Universe within our horizon. We expect that the Universe extends far beyond our horizon, and the Universe beyond our horizon may not have the same properties as the observable Universe within our horizon. - The Big Bang model does not suppose that an explosion occurred at one point that is the center of the Universe. Instead, the entire Universe was once a point, and it has now expanded in size to its current extent. So every point in our Universe now was once in the same location in the past. So, the proper way to consider the location of the "center" is that every single location in the Universe now can be considered to be the center of the Universe equally.
- A very common question is to ask "What is beyond the edge of the Universe or what is the Universe expanding into?" The (unsatisfying) answer to this question is that the Universe encompasses everything and it as a whole is expanding, so there is no edge.
- The Big Bang model does not claim to explain the cause of the initial expansion.

**Links**

[1] http://www.astronomynotes.com

[2] http://www.astronomynotes.com/cosmolgy/s9.htm

[3] http://map.gsfc.nasa.gov/universe/bb_concepts.html