The C routine scalet

The routine scalet first checks for a bouncing universe. The routine then starts calculating a discrete version of eq.  10 which is given in Felten & Issaman [2]:

$$R_n = R_{n-1} + \left(\frac{dR}{d\tau}\right)_{n-1} \Delta\ta... ...c{1}{2} \left(\frac{d^2R}{d\tau^2}\right)_{n-1} (\Delta\tau)^2$$ (13)

where the starting point is R₀ = 1. The coefficient for $\Delta\tau$ is given by eq.  10 and the coefficient for $(\Delta\tau)^2$ can easily be calculated by differentiating eq.  10. The results are:

$$\frac{dR}{d\tau} \left[1+\Omega_M \left(\frac{1}{R}-1\right)+\Omega_\Lambda (R^2-1)\right]^{1/2}$$ = (14)

$$\frac{d^2R}{d\tau^2} 2 R \Omega_\Lambda - \frac{\Omega_M}{2 R^2 \left[1+\Omega_\Lambda (R^2-1)+\Omega_M (\frac{1}{R}-1)\right]^{1/2}}$$ = (15)

The routine scalet makes a difference between negative and positive values for t. For positive values, eq.  13 is used, for negative values the second term on the righthandside of eq.  13 is subtracted from the first term on the right handside. This makes the function also usable for calculating scale factors from t values which range from some negative value to some positive value.