Expansion of the Universe
NGC 4603, a galaxy with Cepheids used to determine distance (14)
The expansion of the universe is an idea that has fascinated astronomers and laymen alike for decades. If the universe started as a point and experienced the Big Bang, it would have to expand to reach its present size. It seems a simple concept on the surface, but as studies progress, we learn more and more how complicated and surreal our universe can be. It has not even been until relatively recently that these ideas could be studied, since they involve such great distances. Technology and theory had to match pace in order for any progress to be made.
An image from COBE showing the large scale smoothness and small scale structure of the Cosmic Microwave Background (16), and a 2dF Galaxy Redshift Survey showing the universe as homogeneous and isotropic on large scales (7)
One of the first and most important assumptions we need to make is that the universe is homogeneous and isotropic. Homogeneous means that the universe is uniformly distributed over space. Isotropic means that whichever way an observer is looking, the universe appears effectively the same. These apply on a large scale, since certainly the universe is neither homogeneous nor isotropic on small scales. We see stars and galaxies in a very distinct distribution in our near neighborhood. However, on large scales, the universe begins to look like a foam, with areas of material and areas of blank space. One ramification of these assumptions is that the Earth is not the center of the universe. Anywhere an observer look, in any direction, he sees basically the same thing, and this shows that there cannot be a center to the universe; otherwise, these ideas would not hold. This allows all of the math and physics to work out reasonably. The cosmic microwave background shows quite well that the universe is isotropic and homogeneous. Extensive surveys have been done that show the CMB to be very smooth until the variations are taken at a very small level, where they begin to differ. The radiation was collected in all directions and at different times, showing that is was constant. It originated from the Big Bang, and now is reaching Earth as a result of expansion in combination with the photon's travel path.
Edwin Hubble, acredited as the discoverer of the universe expansion (16)
Edwin Hubble is the man credited with discovering the expansion of the universe, although there were a number of others on similar paths. Hubble was working at the Carnegie Observatories using a 100in telescope. He measured the distances and redshifts of a number of different galaxies in 1929. His data showed that the galaxies were moving away from our own galaxy. In fact, only the nearest galaxies do not appear to be moving away from us, since they are close enough to be gravitationally affected by the Milky Way. It is important to note that the galaxies are not exactly moving away from us. In fact, it is space that is expanding, and the matter in space is expanding along with it. Things that are gravitationally bound, like a planet or a galaxy, do not experience expansion, but galaxies that are distant from each other can be observed to grow more distant from each other.
It is very hard to measure distances to far away objects. We know how to measure closer objects through simple mathematical methods, but that does not help measure galactic distances. We therefore need some method to obtain larger distances. We call this method the distance ladder, because it allows us to measure larger and larger distances using a stepwise comparison of one method to another. A measurement to one step on the ladder allows the next step on the ladder to be measured, allowing greater and greater distances. Cepheids are a good example because they have easily measurable properties. Cepheids are stars that pulse on a daily to weekly cycle, which is a conveniently measurable amount. Henrietta Leavitt found that the period of pulsation is directly related to the size and brightness of the star. The larger and brighter stars have longer periods. By measuring Cepheids with a comparable distance, she made these measurements that allowed Cepheids to be used as distance measures themselves. Knowing the brightness of the star from its period allows the measurement of distance by comparing the apparent and absolute magnitudes. The formula for distance goes as
.
Redshift is the increase in wavelength of a photon as it moves through space. The higher redshift light has, the longer it has had to travel, and the more distance it has covered. Redshift z follows the formula
, as long as velocity v is much less than the speed of light. Hubble found redshifts in the light from the galaxies he studied. He recognized the redshift of the lines in the spectrum to be enough shifted that the distance was large. The redshift also indicates that the galaxy was moving away from Earth. The redshift directly indicates that other objects are moving away from us with a certain speed that increases with distance.
Hubble plotted his data and from it he obtained the formula v=H0d. This shows that the velocity is directly related to the distance by a factor of H0, the Hubble constant. H0 is determined from the slope of the data. H is dependent on time, such that 
where R is the scale factor. R is related to redshift by R=1/(1+z), so it also changes with time. R is the factor by which a distance increases due to the universe's expansion. Another way to express this is z=H0d/c. There have been a number of efforts to determine what H(t0) is, where t0 is now. Hubble's value was around 500 km/s/Mpc, which is drastically off from the current accepted values. Some errors were included in his value, the main source of which is the Cepheids that he used. They were brighter and of different types than originally thought, which threw off Hubble's estimates. Sandage got a value of 57 km/s/Mpc from the HST key project, whereas Vaucouleurs' value was closer to 100 km/s/Mpc. Friedman got a value of 72 km//s/Mpc. The most commonly used value today is about 71 km/s/Mpc, from the cosmic microwave background.
Hubble diagram showing the velocity versus distance plot, which is effectively linear. H0 describes the slope (7)
Hubble's constant, derived from expansion, can be used to determine the approximate age of the universe. This is called its Hubble time, tH. The Hubble time, tH=1/H0, measures the time since the Big Bang, or the time for the universe to get into its present state. With a value of 3.09*10^17h^-1 s = 9.78*10^9h^-1yr, we can estimate the age of our universe to 10 to 20 billion years.
There are other parameters than H which contribute to the behavior of the universe. One is the density parameter
, which is defined as
, or the ratio of the density to the critical density, where the expansion of the universe is balanced with the gravitational attraction of the matter in the universe. The matter's gravity slows expansion with a drag effect, and this leads to three different possibilities for the motion of the universe. The expansion could be accelerating, constant, or decelerating. In order for acceleration to happen, something needs to overcome the gravitational pull. There have been theories that the energy could come from generating new matter at some small rate, or from dark energy, the energy of empty space, represented by
. It is called the cosmological constant, and was put into equations for mistaken reasons by Einstein. He wanted to keep a steady state universe, but that was thrown out by Hubble's results in 1929. If the expansion and gravity are balanced, then the expansion is constant, and if the gravity is greater than the expansion, it decelerates. This leads to three different cosmological models which are very important for the future of our universe. k is a constant that helps to describe the curvature of the universe, which affects how things move within it. If k>0, then
>1, and it is a closed universe in which the universe's expansion decelerates and will eventually halt and reverse. This could lead to a reverse Big Bang, in which the matter is again condensed into a small area. A flat universe is when k=0 and
=1. In this universe, expansion will continue effectively forever. It slows almost to 0, and halts at t=infinity. The third model is of an open universe, where k<0 and
<1. This corresponds to expansion forever, though it may still decelerate.
Cosmological models of the universe showing closed (orange), flat (green), open with deceleration (blue) and open with acceleration due to dark energy (red) universes (1)
In order to relate all of these parameters into a coherent equation, we use the Friedmann equation. It includes the necessary variables to describe the universe, and goes as 
. If the cosmological constant is 0, then it does not include dark energy. The cosmological constant is coming back into favor however, so it should be kept in mind.
AST 242, Spring 2004, Prof Adam Frank
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