Here we provide you with an exercise to explore the possible universes governed by the Friedmann equation (equation (11.19) in the text). We use a simple applet to integrate the Friedmann equation for a range of models. (Note that this requires Java running on your browser.) On the applet below you can enter a value of Omega (density of the universe), a value of Lambda (the cosmological constant), and select a curvature (positive, zero, or negative). Note: This exercise provides a qualitative feel for the relative behavior of the Friedmann equation with respect to cosmological parameters. It does not provide detail models for comparison with observed cosmological values of omega or hubble time.
1. Begin with the Einstein-deSitter flat standard model. Click "compute" to see the graph of R versus t. In this special model R goes as the 2/3rd power of t.
2. Now experiment with other Lambda=0 models. The graph has been set up to maintain the same scale for all models. Slowly reduce Omega: you will see the curve rise more rapidly than the Einstein-deSitter model. Next, increase Omega above 1. You will see the model curve downward. At large enough Omega the model will recollapse before the right-hand edge of the plot. (If you are having problems remember that you have to select a curvature compatable with your model!)
3. When you have mastered the standard models, try adding a cosmological constant. A negative Lambda will make all models recollapse. Compare a negative Lambda, negative curvature, low density model with the standard closed model. What happens if you compute a high density, positive curvature model but retain a negative Lambda?
4. Now try a positive Lambda. This provides a repulsive force. Manipulate the values of the parameters to get nice looking curves. Can you make a Lemaitre model that has a "hovering" period visible on the graph? You will have to choose your parameters carefully to make the plot fit nicely on the graph.