Running of the scalar spectral index (${\alpha }_s=d{n}_s/d\log k\simeq 16{ϵ}_V{\eta }_V-24{ϵ}_V^2-2{\xi }_V^2$)

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.