Boundaries

As described by Carroll, Press & Turner [1], depending on the value of $\Omega_\Lambda$ the universe may have a turning point in its past, that is, it collapsed from infinite size to a finite radius and is now reexpanding. In general, such bounce cosmologies are ruled out by the mere existence of high redshift quasars and by the cosmic microwave background. The following condition (taken from Carroll, Press & Turner [1]) can be used to detect such a bouncing universe:

$$\Omega_\Lambda \geq 4 \Omega_M \left\{ coss \left[ \frac{1} {... ...left( \frac{1-\Omega_M} {\Omega_M} \right) \right] \right\} ^3$$ (11)

where coss is cosh when $\Omega_M < 1/2$ and cos when $\Omega_M > 1/2$ (for $\Omega_M = 1/2$ the join is analytic). For this reason, the functions described above check this condition and return -1000 if this condition is matched, i.e. the parameters are those of a bouncing universe. However, condition  11 is only usable if w = -1. If the w parameter is used, the condition changes, too. The derivation of  11 can be found in Felten & Isaacman [2]. But this derivation is no longer possible if the w parameter is used, because the Friedman Equations change in a way that no longer allow the derivation in Felten & Isaacman [2]. So the functions only use condition  11 for the case of w = -1. For other choices of w, the user has to make sure himself that this choice of parameters is not a bouncing cosmology, although the functions checks if the argument of the square root in the integral in eq.  2 (or eq.  4 respectively) is negative (and returns -1000 if this is the case).