Scalar spectral index in Hubble slow-roll parameters ($n_s-1=-2ϵ-\eta$)

Differentiating the curvature spectrum at horizon exit givesto first order in Hubble slow-roll parameters.

The tensor spectrum is proportional to $H^2$, but the single-field slow-roll curvature spectrum is proportional to $H^2/{ϵ}_H$. Thus $n_s-1\simeq -2{ϵ}_H-d\ln {ϵ}_H/dN$. A decreasing $H$ gives a red leading tensor tilt, while the evolving scalar conversion factor can change the sign of the scalar tilt.