= Solution
A perturbation of the <inflaton> shifts its local clock: $\delta\phi\simeq\dot\phi\,\delta t$. Regions with different clock shifts reach the end of inflation at different times and hence undergo different amounts of expansion. The <inflaton clock-shift origin of curvature perturbations> gives a local expansion difference of magnitude $H\delta t=H\delta\phi/\dot\phi$. The sign depends on whether $\delta t$ denotes a clock advance or a delay to a fixed end value.
The <comoving curvature perturbation> has amplitude $\mathcal R\simeq H\delta\phi/\dot\phi$. For adiabatic evolution, <superhorizon conservation of single-field comoving curvature> carries this amplitude into the later growing <density contrast>, with the numerical conversion depending on the epoch and the density slice. Thus the approximate density-amplitude statement in the question follows; it does not identify the <density contrast> and $\mathcal R$ in every gauge.
The <frozen scalar power> per $d\log k$ is $(H/2\pi)^2$, so the <slow-roll curvature power spectrum> is
$$
\boxed{\Delta_{\mathcal R}^2(k)=\left(\frac{H^2}{2\pi\dot\phi}\right)^2_{k\text{ at horizon exit}}.}
$$
Consequently $\langle\mathcal R^2\rangle=\int (dk/k)\Delta_{\mathcal R}^2(k)$, giving the requested approximate growing-density spectrum. Both $H$ and $\dot\phi$ are evaluated separately when each <Fourier mode> undergoes <cosmological horizon exit>.
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