 Solution

The ordinary Sachs-Wolfe effect combines intrinsic last-scattering temperature with the gravitational redshift of photons escaping the last-scattering gravitational potential. For adiabatic large-scale perturbations in matter domination, with the potential convention in the question, the combination is $\Theta =\varphi /(3c^2)$, up to the observer monopole. It is distinct from the Integrated Sachs-Wolfe effect caused by a time-varying potential along the later line of sight.

Use the Fourier transform convention in which $P_{\delta }(k)=Ak$ is the matter power spectrum and $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_D(\mathbf{k}-{\mathbf{k}}^{\prime })P_{\delta }(k)$. The Poisson equation becomes $-k^2{\varphi }_{\mathbf{k}}=4\pi G{\bar{\rho }}_0{\delta }_{\mathbf{k}}$, givingWith the matter-dominated critical-density approximation ${\bar{\rho }}_0=3H_0^2/(8\pi G)$,independent of $k$. Thus the Harrison-Zeldovich spectrum gives scale-invariant potential fluctuations and the requested $H_0^4$ dependence. Keeping a general present matter fraction would insert ${\Omega }_m^2$; the answer here follows the stipulated critical matter-dominated normalization. In this Fourier transform convention, $A$ has dimensions of length to the fourth power.