## Solve the Friedmann Equation

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.