= Solution
Consider a logarithmic band of <wavenumbers> around $k\sim R^{-1}$, where $R$ is a comoving scale. The <dimensionless cosmological power spectrum> of the <density contrast> is $\Delta_\delta^2(k)=k^3P_\delta(k)/(2\pi^2)$. For $P_\delta=Ak$ this scales as $k^4$, so the rms density amplitude in such a band scales as $R^{-2}$.
The <cosmological Poisson equation> gives $-k^2\Phi_{\mathbf k}=4\pi Ga^2\bar\rho_m\delta_{\mathbf k}$, with $\Phi$ a dimensionful <peculiar gravitational potential>. Its spectrum and its <potential fluctuations per logarithmic wavenumber> are therefore
$$
P_\Phi(k)=\frac{(4\pi Ga^2\bar\rho_m)^2A}{k^3},
\qquad
\boxed{\Delta_\Phi^2(k)=\frac{(4\pi Ga^2\bar\rho_m)^2A}{2\pi^2},}
$$
independent of $k$. Equivalently, $\Phi_R\sim G\bar\rho_m(aR)^2\delta_R$ is independent of $R$ because $\delta_R\propto R^{-2}$. This is the scale-local meaning of the <Harrison-Zeldovich spectrum>. An uncut spectrum has a divergent total potential <variance>; equal amplitude per logarithmic band does not imply a finite unsmoothed <variance> over all scales. For wavelengths beyond the <Hubble radius>, the statement is formulated using the corresponding conserved primordial curvature mode and its adiabatic matter-era potential, rather than treating the subhorizon Poisson approximation as universally valid.
On angular scales larger than roughly a degree, the <Sachs-Wolfe effect> relates the last-scattering potential to the observed <Cosmic microwave background anisotropy>: $\Delta T/T\simeq\Phi/(3c^2)$ for adiabatic modes in <matter domination>. Thus the observed large-angle amplitude constrains $A$. Scale-invariant potential bands produce a <Sachs-Wolfe plateau>, $\ell(\ell+1)C_\ell$ approximately constant, rather than constant $C_\ell$. Its amplitude and any tilt test the normalization and spectral shape. <Cosmic variance>, <Integrated Sachs-Wolfe effect> and assumptions about adiabatic initial conditions limit this inference; the density spectrum's late-time transfer must also be included when comparing it with galaxy clustering.
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