= Relaxation to the slow-roll attractor
{title2=$\dot\phi=-\frac{V'}{3H}(1-e^{-3H\Delta t})$}
With approximately constant positive $H$ and potential slope $V'$, a homogeneous <scalar field> initially at rest solves $\ddot\phi+3H\dot\phi=-V'$ with the displayed <velocity>. The transient decays on time $1/(3H)$, leaving the <slow-roll approximation> $3H\dot\phi\simeq-V'$. The fractional potential change over a <Hubble time> is of order $M_{\rm Pl}^2(V'/V)^2$. Sustained slow roll normally uses both a small slope parameter and a small absolute curvature parameter $|M_{\rm Pl}^2V''/V|$. Local derivative bounds must hold over the traversed field interval; initial small <kinetic energy> alone is not a global inflation-duration theorem.
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