= Solution
Differentiate the <Friedmann equation> and use cosmological energy conservation:
$$
2H\dot H-2\frac{kc^2}{a^2}H
=\frac{8\pi G}{3}\dot\rho
=-8\pi GH(\rho+P/c^2).
$$
For an expanding universe $H>0$, division by $2H$ and substitution of the <Friedmann equation> give the <Friedmann acceleration equation>
$$
\boxed{\frac{\ddot a}a=\dot H+H^2=-\frac{4\pi G}{3}(\rho+3P/c^2).}
$$
The <strong energy condition> makes $\ddot a\leq0$. Since $aH=\dot a$ and $\dot a>0$,
$$
\frac d{dt}(aH)^2=2\dot a\ddot a\leq0.
$$
Thus the quantity is nonincreasing, with strict decrease when the energy condition is strict. Dividing the <Friedmann equation> by $H^2$ yields
$$
\boxed{\Omega-1=\frac{kc^2}{(aH)^2}.}
$$
For fixed nonzero curvature, the magnitude of $\Omega-1$ grows as expansion decelerates. The observed closeness to one therefore requires much tighter early-time closeness to one in the standard decelerating model: this is the <flatness problem>. Accelerated expansion during inflation reverses this evolution and suppresses the curvature contribution.
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