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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.

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.