29:50 Modern Astronomy
Fall 1999
Lecture 30 ...November 5, 1999
Cosmology II
Last time I talked about describing the universe as a whole within the context of the theory of General Relativity. This consists of studying solutions of the the Einstein equations.
How do we solve the Einstein equation for the universe as a whole? You can't. You have to make some approximations which simplify the mathematics to the point where you can get some solutions to the equations. One of the primary ones is to assume a Friedmann Universe, in which the real universe is replaced by a paté with the same density.
Illustration of idea of Friedmann Universe.
This then leads to a equation for the radius of curvature of the universe
which is solvable (don't get hysterical; we are not going to solve this in this class!)
where R is the radius of curvature, 
is the mean density of matter in the universe,
and k is a parameter which the universe chose and determines the nature of the solutions.
This parameter can be negative, positive, or zero. It determines the nature of the solutions.
Figure 26.12 of text.
The value of k also determines the type of curvature of spacetime. A negative value
corresponds to ``open'' or saddle-like curvature, while a positive value corresponds to
``closed'' curvature like the surface of a sphere.
Figures 26-4 and 26-9.
The value of k, and the correct solution in Figure 26.12, is determined by the
ratio of the density of matter in the universe to the critical density 
.
For the values of the Hubble constant we have been discussing, is about 
kilograms per cubic meter. This corresponds to a mean density of about 6 hydrogen atoms
per cubic meter throughout the universe. By way of contrast, the density of the interstellar
medium, or tenuous gas between the stars, is about 
atoms per cubic meter.
What these results mean is that one can determine the geometry of spacetime, and the ultimate fate of the universe, but measuring the average density of the universe and comparing it with the critical value.
Results for Measuring 
.
Let's imagine drawing a big box centered on the Milky Way and measuring
100 Megaparsecs on a side. The volume of this box is V. Let's count up all the galaxies. We then multiply the number
of galaxies by the mass of each galaxies in stars. Let the mass in stars be 
.
Then the density in stars is 
. We find that 
.
If this is all the matter, the universe is way open; there is not nearly enough
mass to close the universe.
Question: What could we be missing in such a calculation?
We have seen before that most of the mass in large galaxies is in the form of
Dark Matter, perhaps as much as 80 % to 90 % of the mass in a large galaxy. We should then
multiply the above number by 5-10 to get the total known mass in galaxies. We would then
have as an estimate 
. This is still way less than
unity.
The data therefore strongly indicate that there is not enough matter to close the universe. We therefore live in an open universe, and the universe happened only once.
The Big Bang
The equation discussed before, derived from General Relativity, says that if we follow the history of the universe backwards, we come to a time when ``Cosmic Scale Factor'', or ``Radius of Curvature'' of the universe was zero. This happened about 14 billion years ago. At that time the density and temperature of the universe would have been infinite. This event is referred to as the Big Bang.
If we follow the history of the universe since that time, we have the following occurrences.
Transparencies with history of the universe
The picture as I have described it is called Standard Big Bang Cosmology. The next
topic to consider is further evidence that this is really what happened.