Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 1 / iv / Solution

The stationary-point relation is ${\Omega }_{m,0}{a}_0^3/a_{\min }=2{\Omega }_{\Lambda ,0}{a}_{\min }^2$. ConsequentlyTaking the positive root on the expanding branch and eliminating $a_0$ with the result of part (iii),The bracket must be positive for a genuine passage through this minimum expansion speed in a matter-Lambda universe. Negative values mean that this formal value of the scale factor is not accessible on the corresponding physical expanding solution.