Age of the Universe Exercise

Background

Any successful model of the universe should predict an age for the universe that is older than the oldest observed stars. The age of the universe depends on two parameters; the current rate of expansion of the universe, also known as the Hubble constant (H), and the rate at which the expansion is slowing down. The expansion of the universe slows down due to the gravitational attraction of all the matter in the universe. This means that the density of matter in the universe tells you the rate at which the expansion is slowing. Cosmologists usually give the density of matter in terms of the density necessary to just cause the universe to re-collapse (Omega Matter). This means that if (Omega Matter) = 1.0 there is just enough matter for the universe to re-collapse, (Omega Matter) < 1.0 means that there is not enough matter to cause the universe to recollapse, and (Omega Matter) > 1.0 means that there is more than enough matter to re-collapse the universe.

Another possible thing can affect the expansion, and thus the age, of the universe. The theory of general relativity does not exclude the possibility of a "Cosmological Constant" (signified by the Greek letter Lambda), a term in the equations of general relativity that acts to speed up the rate of expansion with time. A non-zero value for Lambda may arise due to the effects of quantum mechanics, which allows for the existence of virtual particles; particles which pop into existence (in particle+anti-particle pairs), and then annihilate one another in a time short enough that they don't violate the laws of quantum mechanics. The effect of a cosmological constant can be thought of as contributing to the value of Omega, that is: (Omega Total)=((Omega Matter)+(Omega Lambda)).

What Are H, (Omega Matter), and (Omega Lambda)?

The best current measurement of the Hubble Constant H is due to a Hubble Space Telescope key project to measure the distance to other galaxies, using Cepheid variable stars. The current value found by this project for H is 70 km/sec/Megaparsec.

By watching how fast galaxies rotate, or how fast galaxies orbit in clusters of galaxies, astronomers can measure the amount of mass in the galaxy or cluster, and thus estimate the total amount of mass in the universe. Current measurements give values of (Omega Matter)=0.3.

Many theoreticians would like (Omega Total) = 1.0 (exactly). Although the universe in the past has felt no particular need to obey the wishes of theoreticians, notice that if the theoreticians are correct, that there are two ways that this could happen:

1. The measured value of (Omega Matter) is way off. Although possible, this seems unlikely, since the measurements have been getting better over the last 20 years, but the estimated value of (Omega Matter) from the data has not risen.
2. Since (Omega Total)=((Omega Matter)+(Omega Lambda)), (Omega Lambda)=0.7 would also give (Omega Matter)=1.0.

Because a non-zero cosmological constant will cause the expansion of the universe to speed up with time, measurements of the distances and recession speeds of distant galaxies can be used to try and measure the value of (Omega Lambda). Groups of astronomers have been trying to do just that, using supernovae in distant galaxies to estimate their distances. These astronomers find that the galaxies are indeed receding faster than would be expected if Lambda = 0 (Although this is a tough measurement to do, so astronomers are still arguing about whether this is correct or not). If the result does stand up, it is consistent with (Omega Lambda) = 0.7.

How Old are the Oldest Stars?

There are two primary ways astronomers estimate the ages of the oldest stars; using the HR diagram of globular star clusters along with models of how long stars of different mass can remain on the main sequence, and observations of the lowest luminosity white dwarf stars, along with models of how white dwarfs cool with time.

• Main Sequence Fitting: The textbook quotes a value of 11 billion years as the oldest age from main sequence fitting. From the latest reviews in the astronomical literature, I find 11.5-12 billion years.
• White Dwarf Cooling The text quotes 10 billion years. The literature values I find are 9.5-10 billion years.

The Exercise

Professor Chris Mihos at CWRU has a nice java applet that lets you put in different values for H, (Omega Matter), and (Omega Lambda), and plots graphs of the age of the universe, the lookback time (i.e. how far back in time are you looking when you observe an object at a given redshift), and scale factor of the universe, all as a function of redshift.

To get to the applet, go to the Dynamical Astronomy JavaLab, (Note you will probably want to open another browser window for the JavaLab, so you can see the graphs and these instructions at the same time). From the main JavaLab page, select applets from the buttons along the side, and then select Cosmo. Important Note: before running the applet, it is a good idea to read the "Background" page, that describes what the applet does, and the "Controls" page, so you have some idea of how you will control the simulation.

1. First take a look at the age of the universe predicted assuming (Omega Total) = 1.0 with the values for the other parameters given above. Enter those under case 1, select "plot age" from the left pull down menu, and click "trace". Note that the current age of the universe on the graph is for redshift = 0, (i.e. the very left side of the graph). What age for the universe is predicted, is it greater than the main sequence and white dwarf cooling estimated ages?
2. Now see what happens when the cosmological constant is zero, but the other values stay the same. For case 2, enter the same H and (Omega Matter) values you had in case 1, but enter 0 for (Omega Lambda), corresponding to no cosmological constant. Click "trace" again. Is the new age estimate younger or older than the stars?
3. Try fixing things up (i.e. getting an age older than the stars) by just varying H, and leaving the Omega parameters alone. Try entering the same values of (Omega Matter) and (Omega Lambda) as in case 2 for case 3, but changing the Hubble Constant to get an older age. What value of H is necessary to match the case 1 age? Do you have to make the value of H larger or smaller to increase the age of the universe?
4. Finally, try making the theoreticians happy (i.e. set (Omega Total) = 1.0), without a cosmological constant. What does that mean for the values of (Omega Matter) and (Omega Lambda)? using case 4, what value of the Hubble Constant is necessary to give an age for the universe older than the oldest stars in this case?

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