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Assume $V>0$ and that its derivatives do not change substantially over the field interval traversed in the estimate. With initially negligible kinetic energy, $H^2\simeq V/(3M_{\mathrm{Pl}}^2)$. The force $-V_{,\varphi }$ changes the field velocity on the Hubble friction time. Keeping $H$ and $V_{,\varphi }$ approximately constant for that local estimate, and starting at rest, solves the field equation explicitly:After one Hubble time,Writing $A=M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/V^2$, the condition $A<1$ keeps this kinetic-to-potential ratio below about $0.31$, comfortably inside the inflationary bound. Integrating once more gives the fractional potential change estimateover the same interval. Thus the characteristic time for an order-one potential change is $H^{-1}/A$, up to constants of order one. The requested local Hubble-timescale criterion isUsing a strict inequality with one as the order-one threshold is conventional; its coefficient is not an exact universal end-of-inflation bound. Much smaller $A$ gives the controlled slow-roll regime and many Hubble times of cosmic inflation if the bounds persist.

After the transient in relaxation to the slow-roll attractor, $3H\dot{\varphi }\simeq -V_{,\varphi }$. Differentiating this approximate relation givesTherefore the standard sufficient conditions for negligible acceleration throughout slow roll are ${ϵ}_V\ll 1$ andThe absolute value matters: large negative curvature also drives rapid evolution. The more precise local acceleration criterion is $|{\eta }_V-{ϵ}_V|\ll 1$, with the usual separate smallness conditions avoiding a tuned cancellation. If a field starts exactly at rest on a nonzero slope, its initial acceleration is not negligible; the displayed exponential transient describes its approach to the attractor.

These are local timescale and slow-roll conditions, rather than a global duration theorem from derivatives at a single point. A smooth potential can have a narrow flat shoulder followed by a steep drop, ending cosmic inflation soon despite small starting derivatives. The conditions must hold over the traversed interval. This is the additional smooth-evolution assumption implicit in the estimate.