= Solution
Use <Newtonian gauge> with scalar <metric perturbations>
$$
ds^2=a^2\bigl[-(1+2\Psi)d\tau^2+(1-2\Phi)d\mathbf x^2\bigr].
$$
Let $\widehat n$ point from the observer toward the emitter, $\Theta=\delta T/T$, and evaluate the unperturbed ray at $\mathbf x(\tau)=(\tau_0-\tau)\widehat n$. In the sudden-decoupling approximation the <Sachs-Wolfe effect> and its linear companions give
$$
\boxed{\Theta(\widehat n)=\bigl[\Theta_0+\Psi-\widehat n\cdot\mathbf v_b\bigr]_*+
\int_{\tau_*}^{\tau_0}(\Phi'+\Psi')\,d\tau-\Psi_0+\widehat n\cdot\mathbf v_0.}
$$
Here $\Theta_0=\delta_\gamma/4$ is the intrinsic <photon monopole>; $\Psi_*$ is the emission <gravitational redshift>; the velocity term is the <Doppler CMB anisotropy> of the baryon-photon plasma; and the integral is the <Integrated Sachs-Wolfe effect> from evolving potentials during travel. The last two local terms are an observer monopole and dipole, removed when discussing the higher measured multipoles. Changing the direction convention reverses both velocity signs. A finite <cosmological visibility function> replaces the sharp emission surface by a weighted source integral.
For scale-invariant primordial potentials, $\mathcal P_\Psi=k^3P_\Psi/(2\pi^2)$ is constant before transfer effects. In <matter domination>, the adiabatic superhorizon relation is $\Theta_0=-2\Psi/3$, so the observable <Sachs-Wolfe combination> is $\Theta_0+\Psi=\Psi/3$. Its amplitude per logarithmic scale is therefore approximately independent of angle. In a flat geometry $\ell\sim kD_*$ and $\theta\sim\ell^{-1}$, and projection gives the <Sachs-Wolfe plateau>
$$
\boxed{\ell(\ell+1)C_\ell\simeq\text{constant}.}
$$
Both the intrinsic and gravitational emission terms have this leading large-angle scaling; they must be combined, not treated as unrelated random sources.
The Euler equation gives $v_b\sim k\tau_*\Psi$ for the growing large-scale matter mode. Thus the Doppler amplitude grows roughly as $\theta^{-1}$ relative to the large-scale potential amplitude, while its logarithmic power grows as $\theta^{-2}$, until acoustic dynamics and damping invalidate that simple estimate. On subhorizon pressureless scales the <Poisson equation> gives $\delta_m\sim(k\tau)^2\Psi$: the corresponding density amplitude has the often-used $\theta^{-2}$ scaling. This is a matter-density estimate, not a universal small-angle photon-temperature law; the actual intrinsic photon source undergoes the oscillations in part(b). On superhorizon scales it instead has the constant-potential behavior just described.
The integrated source vanishes for constant linear potentials in pure <matter domination>. Potential decay during <radiation domination> or late accelerated expansion makes it nonzero. Its broad, slowly evolving sources favor large angular scales and give a potential-sized contribution there; crossings of many structures cancel progressively on small scales. Scale invariance alone fixes no universal power of angle for the integrated term: its time evolution and transfer functions are also required. Consequently the large-angle plateau, velocity-gradient enhancement and subhorizon density-gradient estimate describe distinct physical regimes, rather than one extrapolated power law across the whole spectrum.
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