= Solution
Before decoupling, rapid <Thomson scattering> makes the <photon> distribution nearly isotropic in the local photon-fluid rest frame. A local blackbody <temperature> perturbation changes its <energy density> by $\delta_\gamma=4\delta T/T$. A first-order boost by the fluid <velocity> adds a <temperature> <photon dipole> $\mathbf n\cdot\mathbf v$; it does not generate a <photon quadrupole> until second order in that boost. Thus the <photon brightness perturbation> at instantaneous decoupling is
$$
\Delta_* =\delta_{\gamma *}+4\mathbf n\cdot\mathbf v_*.
$$
Higher angular moments are suppressed by the <photon> mean-free-path to perturbation-length ratio in the <tight-coupling approximation>. The idealized equilibrium assumption neglects those finite-mean-free-path corrections, polarization effects and finite thickness of the last-scattering surface. It is not an assertion that collisionless <photons> remain isotropic after decoupling. The distinction between a fluid treatment and the subsequent hierarchy is discussed in https://arxiv.org/abs/astro-ph/9506072[the primary synchronous and Newtonian perturbation treatment].
Let $\mu=\hat{\mathbf k}\cdot\mathbf n$ and $W(\tau)=e^{ik\mu(\tau-\tau_0)}$. Multiplication of the <synchronous photon brightness equation> by its <integrating factor> gives
$$
\left(e^{ik\mu\tau}\Delta\right)'
=-2e^{ik\mu\tau}h'_{ij}n^in^j.
$$
Integration from $\tau_*=\tau_{\rm dec}$ to $\tau_0$ and division by four yield
$$
\boxed{\Theta(\mathbf k,\mu,\tau_0)
=\left(\frac{\delta_{\gamma *}}4+\mathbf n\cdot\mathbf v_*\right)
e^{-ik\mu(\tau_0-\tau_*)}
-\frac12\int_{\tau_*}^{\tau_0}W(\tau)h'_{ij}n^in^j\,d\tau,
\qquad\Theta=\frac{\Delta T}{T}.}
$$
For the real-space observation point $\mathbf x_0$, the corresponding background ray is $\mathbf x(\tau)=\mathbf x_0-\mathbf n(\tau_0-\tau)$. Fourier inversion gives the <synchronous Sachs-Wolfe line-of-sight formula>
$$
\boxed{\Theta(\mathbf x_0,\mathbf n,\tau_0)
=\frac14\delta_\gamma(\mathbf x_*,\tau_*)
+\mathbf n\cdot\mathbf v(\mathbf x_*,\tau_*)
-\frac12\int_{\tau_*}^{\tau_0}
h'_{ij}(\mathbf x(\tau),\tau)n^in^j\,d\tau,
\quad\mathbf x_*=\mathbf x(\tau_*).}
$$
Every source is evaluated at its appropriate point on the ray, rather than at the observation position. Here $\mathbf n$ is <photon> propagation direction, opposite to the outward observer-to-source sky direction; changing that convention changes the written sign of the Doppler projection consistently.
The first term is the intrinsic <photon> <temperature> at last scattering. On wavelengths larger than the <sound horizon>, it retains primordial adiabatic information; inside that horizon the <photon-baryon fluid> has acoustic <energy density> oscillations, producing the acoustic <temperature> peaks. The second term is the <Doppler effect> from the fluid's bulk motion at emission. A regular long-wavelength <velocity> is gradient-suppressed; the Doppler contribution is important near and below the acoustic horizon, with its acoustic phase displaced from that of the <energy density> oscillation.
The integral is the gravitational energy shift produced by the changing spatial <metric tensor> along the <photon> path. In <synchronous gauge> it includes endpoint gravitational redshifts as well as evolving-potential effects. It must not be identified solely with the <Integrated Sachs-Wolfe effect>: in <Newtonian gauge> the same total separates into the ordinary last-scattering <Sachs-Wolfe effect>, local observer terms, and an integral of time-varying potentials. The ordinary contribution is important on scales outside the <sound horizon> at decoupling. A late integrated contribution is strongest on large angular scales when dark energy or curvature changes the potentials; evolution around <matter-radiation equality> also contributes near the first acoustic scales.
The relevant comoving sound scale is $r_s(\tau_*)=\int^{\tau_*}c_s(\tau)d\tau$, while the projection distance is $D_*\simeq\tau_0-\tau_*$. Modes project roughly to <photon temperature multipole> $\ell\sim kD_*$, and the acoustic scale is $\ell\sim D_*/r_s$, with successive peak spacing of order $\pi D_*/r_s$. At much shorter wavelengths <Silk damping> and the finite last-scattering thickness suppress <energy density> and <velocity> anisotropies; the instantaneous-decoupling approximation alone does not model that cutoff. These scale distinctions explain why the three terms cannot be assigned one common range of important wavelengths.
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