Inflaton clock-shift origin of curvature perturbations ($\mathcal{R}=H\delta {\varphi }_{\mathrm{flat}}/\dot{\varphi }$)

On a single-field attractor, a field fluctuation changes the local point along the background trajectory. The corresponding time shift has magnitude $|\delta \varphi /\dot{\varphi }|$, and local expansion converts it into curvature of magnitude $H|\delta \varphi /\dot{\varphi }|$. On spatially flat slices a standard sign convention gives $\mathcal{R}=H\delta {\varphi }_{\mathrm{flat}}/\dot{\varphi }$. The vacuum mode amplitude $H/(2\pi )$ then determines the scalar curvature spectrum.