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Apply the horizon-exit derivative from part (d) to the potential slow-roll parameter and the second potential slow-roll parameter. The potential slow-roll flow equations arewhere the third potential slow-roll parameter is ${\xi }_V^2=M_{\mathrm{Pl}}^4{V}^{\prime }{V}^{\prime \prime \prime }/V^2$. For example, differentiating ${\eta }_V$ gives $-M_{\mathrm{Pl}}^4(V^{\prime }/V)(V^{\prime \prime \prime }/V-V^{\prime \prime }{V}^{\prime }/V^2)$, which is the second expression. Differentiating $n_s-1\simeq -6{ϵ}_V+2{\eta }_V$ now gives the running of the scalar spectral indexThis is the leading second-order slow-roll running. The notation ${\xi }_V^2$ denotes a signed derivative combination and does not force its value to be nonnegative.

The running describes the scale dependence of the scalar spectral index. Differentiating $n_s-1\simeq -6{ϵ}_V+2{\eta }_V$ with the potential slow-roll flow equations gives the displayed second-order slow-roll result. It includes the third potential slow-roll parameter and is generally smaller than the leading spectral tilt.