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

On an expanding branch with $\dot{a}>0$, minimizing $\dot{a}$ is equivalent to minimizing its square. Differentiating the preceding Friedmann equation with respect to $a$ givesFor positive matter density and a positive cosmological constant, it vanishes uniquely atThe second derivative is $H_0^2[2{\Omega }_{m,0}{a}_0^3/a^3+2{\Omega }_{\Lambda ,0}]>0$. Thus this is the minimum expansion speed in a matter-Lambda universe. The matter and vacuum densities satisfy ${\rho }_m=2{\rho }_{\Lambda }$ there, precisely the transition from deceleration to acceleration in the Friedmann acceleration equation.

The conclusion concerns an expanding branch that actually reaches this value of $a$. A sufficiently closed model can recollapse before reaching it; positive $\Lambda$ alone does not rule that out. In flat or open models with the stated positive densities the minimum is reached. The degenerate zero-speed limiting case requires separate treatment and does not describe a finite-time passage through a positive-speed minimum.