= Solution
The <flatness problem> is the extreme closeness to spatial flatness required in the early universe if ordinary decelerating expansion is to produce the observed near-flat universe. From the <Friedmann equation>, with $c=1$ and curvature constant $k$,
$$
H^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2},\qquad \Omega-1=\frac{k}{a^2H^2}=\frac{k}{\dot a^2}.
$$
During decelerating expansion $|\Omega-1|$ grows. The <Friedmann acceleration equation> gives $\ddot a/a=-4\pi G(\rho+3P)/3$. Thus $\rho+3P<0$ makes an expanding universe accelerate, increases $\dot a$ and suppresses $|\Omega-1|$: this is the <inflationary solution of the flatness problem>.
The <slow-roll approximation> neglects $\dot\phi^2/2$ compared with $V$ and $\ddot\phi$ compared with $3H\dot\phi$. For the positive-field branch, taking $m>0$,
$$
H\simeq\frac{m\phi}{\sqrt3},\qquad 3H\dot\phi\simeq-2m^2\phi,\qquad \dot\phi\simeq-\frac{2m}{\sqrt3}.
$$
Integrating the field equation and $\dot a/a=H$ gives \b[the slow-roll solution]
$$
\boxed{\phi(t)=\phi_i-\frac{2mt}{\sqrt3},\qquad a(t)=a_i\exp\left(\frac{m\phi_i t}{\sqrt3}-\frac{m^2t^2}{3}\right)=a_i\exp\frac{\phi_i^2-\phi(t)^2}{4}.}
$$
The <potential slow-roll parameters> are $\epsilon_V=\eta_V=2/\phi^2$. They cease to be small when $\phi$ is of order $1$ in these units; the customary endpoint $\epsilon_V=1$ is \b[$\boxed{\phi_{\rm end}\simeq\sqrt2}$]. Equating the approximate kinetic and potential energies instead gives $\sqrt{2/3}$, another order-one indicator, not a precise endpoint of an approximation that is already failing.
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