This is the last chapter of the course, we will study the universe as a whole. We want to answer questions like: When and how the universe began? When and how it will end? What is ``outside'' the universe? The subject concerned is cosmology.


Cosmological Principles
The three cosmological principles are the basic assumptions about the universe derived from observations. With these assumptions, we can build a simple model on the universe. We do not expect these assumptions are fully satisfied in reality, but by comparing the model and the real universe, we can learn how and why the real one differs.

Universality: This is the most basic assumption, some books even do not include it as one of the principles. We assume that the physical laws we discovered on or near the Earth can be applied to anywhere in the universe. Would it be false, we can talk no more about physical astronomy. Studying the properties of stars and galaxies far away confirms this assumption.

Homogeneity: The principle of homogeneity says that the distribution of matter does not vary with position. This is obviously wrong if we look at the local conditions. Most of the mass of our solar system concentrates at the Sun; most stars in our Galaxy are on the galactic plane. Then, in what sense do we take this assumption?

Our Milky Way Galaxy is in a cluster of galaxies, called the Local Group. It contains other galaxies, such as M31, LMC and SMC. Typically, a cluster of galaxies contains 1000 galaxies within a million parsecs. In a larger scale, clusters group together to form superclusters. It contains about 100 clusters in 100 Mpc (mega parsec, a million parsecs). The Local Group and some nearby clusters, like the Virgo Cluster, form a supercluster called the Local Supercluster.

In an even larger scale, recent galaxy survey shows some large scale structures such as the Great Wall and the Voids. Can we conclude that our universe is inhomogeneous? Though more and more evidence points to the existence of the large scale structures, the answer is no. The reason is that the distribution of the largest structures may be uniform. Even if our universe is inhomogeneous, the homogeneity principle can still serve us as simplified model to work with.

Isotropy: Principle of isotropy says that the universe looks roughly the same in every direction. Isotropy is independent of homogeneity. The universe can be homogeneous but non-isotropic or vice versa.


Cosmic Microwave Background Radiation
In 1965, a great discovery was made. Microwave radiation was detected from all directions in the sky. Later on, scientists realized that the origin of this radiation is cosmological. The spectrum matches with that emitted by a blackbody of temperature 3K. After making the appropriate correction, the radiation is homogeneous and isotropic to a very high precision. We believe this background radiation was generated when the universe was very young. Thus, the cosmological principles were satisfied at the beginning.

Click to enlarge it
Courtesy STScI.


Olbers's Paradox
Don't let the name intimidate you. This paradox rised from the simple question, ``Why the night sky is dark?'' If the universe is infinite in size and the distribution of the star is uniform, then no matter which direction we look, we should see a star if we look far enough, and the night sky should be as bright as the day.

There are many simple but wrong answers to this question. To find out the correct resolution, note that we have to look far away to find a star in most directions. When we look at something far away, we are in fact looking back in time since it takes time for light to travel from the object to us. As a result, we would possibly see back to a time when stars were not formed yet. Then, the sky will be dark at that direction. Here we implicitly use the fact that the universe has a finite age, but how do we know?


Cosmological Redshift and Hubble's Law
We find that most galaxies show cosmological redshift, which means the detected radiation from the galaxies is shifted to longer wavelengths.

We measure the redshift by the following parameter,

z = difference in wavelengths / original wavelength .
For example, if we find that the spectral line of the radiation from a galaxy is of wavelength 515nm, the same spectral line measured on Earth has wavelength 500nm, then the redshift z = 0.03.

The redshift is due to Doppler's effect. Thus, a galaxy having large z means that it is receding from us. If the receding velocity is much less than the speed of light, we can calculate it by the following formula

v = z c
where v is the receding velocity and c is the speed of light. Continuing the above example, the galaxy is moving at a speed of about 9000km/s away from us, given the speed of light is 3x108m/s.

If the motions of the galaxies are random, then we would expect that about half of them will recede from us while half of them will be approaching us. However, Hubble found that not only most of them are receding from us, but also the receding velocity is directly proportional to the distance away from us

v = H d
where H is the proportional constant, Hubble's constant. This is known as the Hubble's Law. This means galaxies further away will move away faster. One could argue that the Hubble's constant is the most difficult parameter to measure in astronomy. Different astronomers using different methods have found different values of H. It ranges from 50km/s/Mpc to about 100km/s/Mpc, with the most accepted value around 60km/s/Mpc. If we know the value of the Hubble's constant, we can determine the distance of a galaxy by means of Hubble's Law. In the example above, the galaxy is about 150Mpc from us.


Big Bang Theory
The sections above are about some observational facts of the universe. Now, we discuss the most widely accepted evolution model of the universe, the Big Bang Theory. It is the simplest theory that agrees with the cosmological principles and most observational facts in our universe. It states that when the age of the universe was about just 1ms, the universe was very small, with very high density and temperature (about 1012K). From that time onward, the universe expanded and its temperature decreased. The cosmic microwave background at 3K is what was left.

In certain sense, the expansion is similar to the baking of bread. The size of the bread increases and the distances between the raisins also increase. If you sit on one of the raisins, you will see that all the raisins are receding from you. Besides, the farther away a raisin locates, the faster it recedes.

It also explains the cosmological redshift. If the galaxy is far away from us, the light it radiates takes a longer time to reach us. During the time of traveling, the universe expands and stretches out the wavelength of the photons.

The Big Bang Theory can tell us the age of the universe. We know how fast the galaxies are receding from us. So, if we run the movie of the universe backward, we can calculate how long it takes for the galaxies to get back together. From this, the age of the universe is about 15 billion years (1.5 x 1010 yr).


Evolution and the Future of the Universe
How the hot, dense and small universe at the beginning evolves to what we have today is a large subject. We still do not know how the large scale structures and galaxies were formed. But we believe during the first million years, the universe was full of photons, much more than the matter. After a million years or so, matter dominates and the large scale structures and galaxies began to form.

Mass density is the physical quantity determining the future of the universe. There is a value called the critical density. If the mass density is larger than the critical density, then the universe will eventually stop its expansion and start to collapse. We call it a closed universe. If the density is smaller, the universe will expand forever. It is an open universe. We say the universe is flat if its density is equal to the critical density. A flat universe is also open.

The closed universe is similar to the surface of a sphere, the size is finite but there is no boundary. The size of an open universe is infinite and its geometry will be similar to a hyperbolic surface. The geometry of the universe also affects the sum of angles of a triangle, but we will not dive into this issue.

The density of the universe is still unknown. One of the major problems is the existence of dark matter. The estimation of the amount of dark matter is very difficult, mainly because we cannot see them. The gravitational lensing effect will help us a bit. Another problem is whether neutrinos have non-zero mass. Since there are an extremely large amount of neutrinos in the universe, they will contribute a lot to the density even if they have a very small mass.

To summarize, we do not know exactly what kind of universe we are living in although latest observations tend to suggest that our universe is open. But more researches have still to be done.