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# The Age of the Universe – Worksheets

Warm-up questions: These questions should be done before you start the activity. Your instructor may go through them as a class, in which case they will not be graded, but they wil be very useful for answering questions in the activity, so make sure you understand them.

1. What does the Hubble constant measure?

2. Compare 2 models at same size (i.e. d is the same for both models.) If v is greater in model A than in model B, will H0 be greater in model A, model B, or the same for both models?

3. For the same two models, will model A be older, or model B, or will they both be the same ag (reminder, the age is 1/H0)?

4. What is Omega Lambda a measure of?

5. Compare 2 models at same size (i.e. d is the same for both models) but one has dark energy. Will the dark energy cause v to increase over time, decrease, or not affect v?

6. If v increases over time, and we measure a value of H0 for now, would H0 have been larger, smaller, or the same in the past?

7. Does having dark energy make the universe appear older, younger, or leave the age unaffected?

## The “Standard” Cosmology

If we look at Hubble’s Law (v=H0 d) we notice that one over the Hubble constant H0 is just distance divided by velocity, which is also a time. Specifically this is the time it would take for any two objects in the universe to move a distance d from each other at an expansion velocity v. This should give us the age of the universe. If you have previously done the Hubble Law lab you have already shown how inverting Hubble’s constant gives the age of the universe. In this lab, you can use the Applet to find the age of the universe.

1. Chose a reasonable value of the Hubble’s constant and enter this under Case 1 in the Cosmo Applet. Set ΩM = 0.0000001 (as close as you can get to 0) . The age of the Universe can be found by showing Plot Age and looking at where the line intersects Age at a redshift of z=0 (now). What age do you find?

2. Calculating the age of the universe just using 1/ H0 (as we did in question 1) assumes that the matter in the universe does not affect the expansion and therefore the age of the universe. Why would the fact that there is matter in the Universe change the age you would get from inverting Hubble’s constant (Hint: H0 = the current rate of expansion, is constant?)?

3. Since we know there is matter in the universe, we know that the actual age of the universe is not just 1/ H0.Do you think the universe would be younger or older than the age found using 1/ H0? Explain your reasoning.

Now test your prediction from question 3.

For Case 1-5 input the same value of H0 and change the value of ΩM between 0.01 and 1 (you can leave the value of ΩL as zero). Record your observations in Table 1.

 Case # H0(km/s/Mpc) Omega Matter Age (Gyr) Case 1 Case 2 Case 3 Case 4 Case 5
1. How does the age of the universe change, as you increase the density of matter in the Universe?

2. Change the Applet to show Plot Size. This is showing you a quantity called the “scale factor” which is a measure of the “size” of the universe. Even if the universe is infinite we can use the scale factor to describe how much the universe has expanded or will expand in the future, compared to the current universe. “r” is the scale factor at any time and “r0” is the scale factor now. Based on the scale factor of the universe, in which Case is the universe growing the fastest? Does this make sense? Explain.

## Open or Closed Universe

Recall that Omega is ratio of the current density of the universe to the critical density that determines whether the universe will expand forever or eventually collapse. Densities of the universe above the critical density will result in a Closed Universe, which will eventually stop expanding. If the density is less than the critical density the Universe will continue to expand forever and is known as an Open Universe. The critical case, where the density of the universe exactly equals that critical density, the Universe will continue to expand forever, but just barely. This is the Flat Universe.

1. Choose a value of the Hubble Constant and three values of ΩM that represent a Closed, Open, and Flat universe. Display these in your Cosmo Applet window and switch to the “Plot Size” graph. Pick a value of ΩM for a Closed Universe that allows you to see when it will collapse again (you may need to experiment with different values).

Sketch out what this plot looks like. Label the axes and the lines representing the 3 different cosmologies.

## Observed H0 and Omega Matter

Now set all of the values of ΩM to the current best estimate: ΩM = 0.3. Change the values of H0 to fall between 50 and 100 km/s/Mpc. Record the results in Table 2.

 Case # H0(km/s/Mpc) Omega Matter Age (Gyr) Case 1 Case 2 Case 3 Case 4 Case 5
1. How does changing the value of H0 change the age of the universe?

2. Hubble Key Project results using Cepheid variable star distances suggest that the best estimate of H0 is 71 km/s/Mpc. Observations of clusters of galaxies suggest that ΩM = 0.3. Given this cosmology, what is the age of the universe?

3. In the 1980s, Globular cluster age estimates fell in the range of 16 to 20 Gyrs. Compare this age to the age of the universe you found in the previous question. Do they agree? Why is this a problem? If you assume that the globular cluster age estimates are correct, how could you change your model to “fix” this problem (what would you have to do to the values of H0 and/or ΩM)?

4. Recent work to improve models of stellar evolution, understanding of globular cluster formation and distances to globular clusters have found a best fit for the oldest globular clusters in the Galaxy to be 12.6 Gyrs. Compare this age of oldest globular cluster stars to the age of the universe you found using H0=71 and ΩM = 0.3.Have recent estimates of globular cluster ages fixed the discrepancy found in the 1980s?

## Modern Values of Omega Matter and Omega Lambda

Both the High-Z Supernova Search and the Supernova Cosmology Project (international collaborations of astronomers) found that the expansion of universe is in fact accelerating, rather than simply decelerating due to the attraction of gravity. To account for this acceleration factor, the cosmological constant, Lambda, has been introduced to our equations that describe the Universe. For H0=71 and ΩM= 0.3, try different values of ΩL and record the resulting ages in Table 3.

### Table 3 - Effect of Omega Lambda

 Case # H0(km/s/Mpc) Omega Matter Omega Lambda Age (Gyr) Case 1 Case 2 Case 3 Case 4 Case 5
1. How does adding a non-zero cosmological constant effect the age of the universe? Explain why you would expect this, given that the cosmological constant was added to the model to explain the acceleration of the universe. (Hint: If the expansion of the universe is accelerating, then it must have been expanding at a slower rate in the past.)

2. Try adjusting the cosmological constant until the age of the universe agrees with the best estimate of the age of the oldest globular cluster (12.6 Gyrs). Assuming that these oldest stars formed soon after the universe began, what limits does this put on the value of ΩL?

3. Observations and theory suggest that the universe is actually flat (ΩTOTAL = 1.0). This is consistent with a value of ΩL of 0.7 if we have found all the matter (which yields ΩM = 0.3).What is the current age of the universe for this model with a value of H0=71?

4. For a quasar with a redshift of z=6, how old was the universe when the light left that quasar, given the parameters from the previous question? What is the look-back time for that quasar?

## Do we need the cosmological constant?

1. It may seem a little strange to add this idea of a cosmological constant to our models to explain how the universe works.  Observations suggest that the universe is flat (ΩTOTAL = 1.0). Assuming that this is correct, and that the value of Hubble’s constant is 71 km/s/Mpc, what would be the consequences if we decided that there was no cosmological constant and constrained our models to just having ΩM = 1.0? What would the age of the universe be?

2. Observations with the HST of globular cluster M4 give an age for that cluster of 12.7 +-0.7 Gyrs. If star formation in our galaxy less than 1 billion years after the Big Bang, is the age of M4 consistent with a cosmological model that has the cosmological constant Lambda=0, even if we assume there is more matter out there, that we just haven’t found yet? Is it consistent with the age of the universe if ΩL = 0.7?

3. Observations of elliptical galaxies at redshifts of about z~=1.5 suggest that at that age of these galaxies are greater that or equal to 3 billion years. What different models of the universe (i.e., values of H0, ΩM, and ΩL) are ruled out by this observation? Remember that galaxies were finished forming about 1-3 Gyrs after the formation of the universe. You may find it useful to plot a point at z=1.5 and age=3 Gyrs.

Updated: 7/9/10 by SAM.

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