Potential slow-roll flow equations ($d{ϵ}_V/d\log k=4{ϵ}_V^2-2{ϵ}_V{\eta }_V,d{\eta }_V/d\log k=2{ϵ}_V{\eta }_V-{\xi }_V^2$)

At horizon exit during single-field slow-roll inflation, $d/d\log k\simeq -M_{\mathrm{Pl}}^2(V^{\prime }/V)d/d\Phi$. Applying this operator to the potential slow-roll parameter and the second potential slow-roll parameter yields the displayed leading-order flow equations. Corrections to the horizon-exit derivative contribute only at higher slow-roll order here.