Slow-roll curvature power spectrum (${\Delta }_{\mathcal{R}}^2$)

At leading order in canonical single-field slow roll and evaluated at horizon exit $k=aH$,

At a fixed horizon-exit e-fold count and with the later cosmology fixed, rescaling a canonical single-field inflationary potential by a positive constant leaves its potential slow-roll parameters and field solution unchanged. The slow-roll curvature power spectrum is proportional to $V/{ϵ}_V$, so the matter power spectrum scales by the same factor while its leading tilt and transfer-function shape remain unchanged. A changed specified reheating history can also shift the horizon-exit count assigned to a fixed physical wavenumber.

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