Comoving curvature perturbation ( $R$ )

The comoving curvature perturbation is the scalar curvature perturbation on hypersurfaces orthogonal to the total matter flow. For single-field adiabatic inflation it is conserved on superhorizon scales.

Slow-roll curvature power spectrum ( $Δ_{R}^{2}$ )

At leading order in canonical single-field slow roll and evaluated at horizon exit

k = a H

Δ_{R}^{2} (k) = \frac{1}{2 ϵ M _{Pl}^{2}} (\frac{H}{2 π})^{2} .

Differentiating the curvature spectrum at horizon exit gives

n_{s} - 1 = \frac{d l o g Δ _{R}^{2}}{d l o g k} = - 2 ϵ - η

to first order in Hubble slow-roll parameters.

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