= Solution
For $\Lambda=0$, use the instantaneous <cosmological density parameter>
$$
\Omega(a)=\frac{8\pi G(\rho_M+\rho_R)}{3H^2}
=\frac{H_0^2(\Omega_{R0}a^{-4}+\Omega_{M0}a^{-3})}{H^2}.
$$
Insert the <Friedmann equation> from the preceding part and multiply numerator and denominator by $a^4$:
$$
\boxed{\Omega(a)=
\frac{\Omega_{R0}+\Omega_{M0}a}
{\Omega_{R0}+\Omega_{M0}a+(1-\Omega_0)a^2},\qquad
\Omega(a)-1=
\frac{(\Omega_0-1)a^2}
{\Omega_{R0}+\Omega_{M0}a+(1-\Omega_0)a^2}.}
$$
This holds on an expanding or contracting branch with $H\ne0$; in a recollapsing model the density parameter diverges at turnaround.
During <radiation domination>, while curvature is small, $|\Omega-1|\simeq|\Omega_0-1|a^2/\Omega_{R0}$. During <matter domination> the corresponding behavior is $|\Omega-1|\simeq|\Omega_0-1|a/\Omega_{M0}$. Thus the exactly flat solution $\Omega=1$ is unstable against relative curvature during decelerating expansion: a small departure grows rather than being dynamically driven to zero. The <flatness problem> is that a universe with modest present curvature must have had an extremely tiny initial curvature-to-density ratio at an early radiation epoch. The fact that every such solution tends to $\Omega=1$ when extrapolated to $a\to0$ does not supply a mechanism selecting that small departure at a specified finite initial epoch. An early period of <cosmic inflation> can instead shrink the <comoving Hubble radius> and suppress $|\Omega-1|=|k|/(aH)^2$.
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