29:50 Modern Astronomy
Fall 2002
Lecture 32 ...December 4, 2002
Cosmology II

Watch the skies! Watch the skies!
Geminid meteors this month.

(1) Last time we discussed Hubble's Law and its consequences.

The precise value of the Hubble constant is currently believed to be about 70 kilometers per second per Megaparsec.
(a) We are living in an expanding universe. (b) Hubble's constant itself is a clue to how long ago this happened.

(2) The fact that the universe is expanding means that it must have begun in a state of infinite density and temperature and has been expanding ever since. This titanic event of long ago is referred to as the Big Bang.

(3) Let's look in more detail at exactly what the timescale is telling us. For this, we need to look at the physics of the universe. As we will see in the next few lectures, this is an area of astronomy where the standard story has changed in an important way in the last few years.

(4) There is no center to the universal expansion. This is not intuitive, but that's the way it is! Every observer in the universe would see galaxies expanding away from it in all directions. Sort of an explanation is given later.

If Cosmology is to be a branch of physical science, there must be an underlying mathematical structure with quantitatively testable predictions.

(5) There are two things going on in the universal expansion. (1) Galaxies are flying apart due to the universal expansion and (2) the force of Gravity is acting to pull them back together again. The story of the universe is the story of a competition between the universal expansion and gravity.

The best theory of gravity is General Relativity, a theory previously encountered to describe Black Holes. There are two components of GR. (1) Dynamics of systems takes place in a Four Dimensional Spacetime. (2) Mass induces warping or curvature of spacetime.

In some physical contexts, this curvature becomes pronounced, and space and time become inextricably linked or ``coupled''. An example we have seen is Black Holes. A second important example is the universe as a whole.

When we look on cosmological scales, on distances , you cannot just view the universe as three dimensional, with time as a parameter. The universe exists as 4D entity with strong curvature effects. You can't even qualitatively understand what is going on if you stick to a 3D view.

An analogy The surface of the Earth as a 2D surface embedded in a 3D space. Locally, it looks purely 2D. On scales of the size of the Earth, the 3D nature is unavoidable.

Let's talk about Cosmology from the viewpoint of General Relativity.

Mass warping of spacetime (curvature). This is described by Einstein's equation.

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.

The approximations which have been made to allow solutions to Einstein's equation are:

• The real universe can be approximated by one in which the matter is smoothed out to a uniform density .
• The universe is isotropic, meaning it is the same in any direction.
• The universe is homogeneous, meaning that at a given time, it looks the same anywhere in the universe.

(6) When you make these simplifications, the Einstein equation gives you an equation, which you can solve, for R(t), the ``cosmic scale factor'', which tells you essentially how big the universe is. R(t) means it depends on (`` is a function of'') time. The solutions you get are plotted in Figure 26.12.

(7) 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 is about 10 Hydrogen atoms per cubic meter, on the average.

(8) Looking at Figure 26.12, we can categorize the universe as being on one of the curves labeled ``open'', ``flat'', or ``closed''. One of the big questions in cosmology is determining which of these (if any) represents the real universe.

What the above discussion means 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.

(9) After inventing General Relativity, Einstein noticed the fact that the solutions had the form shown in Figure 26.12. At the time, there was no evidence for an expansion of the universe, so Einstein felt that he had omitted something which would allow solutions with R= constant. The equations were doctored up by inserting a quantity called the Cosmological Constant, which was arbitrary, but for certain values would have R= constant. In the discussion of cosmology, the symbol is used for the cosmological constant. A couple of years later, Hubble announced his discovery that the universe was expanding, and Einstein said that introducing it was the ``worst mistake of my career''. However, many people have pointed out that we do not know for sure that .

(10) Let's return to the question of the meaning of the timescale . The Hubble constant is related to the slope of the graph of R(t). If the slope of R(t) was always the same as at present, all the way back to the Big Bang, T would be the age of the universe.

(11) Look at Figure 26-13 from the book. Note that T is the oldest the universe could be, if the Friedmann Universe is the correct description of things. The oldest objects you know of had better be less than this age, or you have a contradiction.

(12) The discussion could go in a number of directions at this point, but we will go back and talk about conditions at the very beginning, i.e. the Big Bang. The fact that this occurred is true regardless of whether we live in an open or closed universe.

(13) 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

• First few seconds: really weird stuff
• First three minutes: whole universe hot and dense as the center of the Sun. Nuclear reactions going on everywhere.
• 700,000 years after BB. Gas has cooled to point where protons and electrons recombine, making universe transparent. Decoupling of radiation and matter.
• Few hundred million years: Formation of ghostly protogalaxies.
• One billion years after BB: Birth of the Quasars
• Five Billion Years after BB: Galaxies as they are today
• 14 billion years after BB: Commercials, politicians, New Age music

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.

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