Fortran Program Using a Second Order WKB Approximation This was used to generate all of the curves in all of these figures except for one curve in Figure 8. It uses a set of routines from the book Numerical Recipies by Press et. al. as the differential equation solver. The user must supply these or replace them with routines for a different differential equation solver.

Fortran Program Using a Zeorth Order WKB Approximation This was used to generate one of the curves in Figure 8. It uses a set of routines from the book Numerical Recipies by Press et. al. as the differential equation solver. The user must supply these or replace them with routines for a different differential equation solver.

Figure 6: File modes.dat read in by the program. Data for k = 1, for k = 10, for k = 100, and for k = 1000.

Figure 7: I no longer have the modes.dat file for this data but it is similar to the one for Figure 6. Data for k = 1, for k = 10, for k = 100, and for k = 1000. Data file used to get the large u behavior for k = 1000.

Figure 8: File modes.dat read in by the program. Data for k = 10 for the zeroth order WKB approximation and for the second order WKB approximation.

Figure 9: File modes.dat read in by the program for m = 1, file modes.dat read in by the program for m = 3, and file modes.dat read in by the program for m = 5. Data for k = 1000 for m = 1, for m = 3, and for m = 5.

Figure 11: I no longer have the original modes.dat file. However, I did the calculation with modes up to k = 8000. Data file.

Figure 12: I no longer have the original modes.dat file. However, I did the calculation with modes up to k = 2000. Data file. For the plot I multiplied the energy density of the particles which is the 5th column by the 4th power of the scale factor. The first column is the dimensionless time u and the scale factor is Cosh[u]/H.

Figure 13: I no longer have the original modes.dat files. However, I did the calculations with modes up to k = 2000. Data file for ui = 5. Data file for ui = 3.