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The conventional vertical variable is $D_{\ell }=\ell (\ell +1)C_{\ell }/(2\pi )$ in $\mu {\mathrm{K}}^2$, plotted against the dimensionless angular multipole $\ell$. The Sachs-Wolfe effect supplies a nearly flat large-angle plateau for $\ell \lesssim 30$. The total spectrum has its first acoustic peak near $\ell \simeq 220$ with $D_{\ell }$ of order $5\times 10^3\mu {\mathrm{K}}^2$, followed by further Cosmic microwave background acoustic peaks; the Doppler CMB anisotropy is phase shifted relative to the photon-density oscillation. At $\ell \gtrsim 1000$, Cosmic microwave background diffusion damping lets photons random-walk across perturbations during recombination and produces the rapidly falling damping tail.

Along the unperturbed photon path, combine the temperature and gravitational-redshift terms and use ${\mathcal{T}}^{\prime }=-\Gamma$:The integrating factor is $e^{-\mathcal{T}}$, and $g=\Gamma e^{-\mathcal{T}}$. Neglecting the exponentially hidden initial boundary term gives the Cosmic microwave background line-of-sight solutionFor instantaneous recombination, $g(\tau )=\delta (\tau -{\tau }_{\mathrm{dec}})$, and the observer potential contributes only an unobservable monopole. ThereforeThe first two terms form the ordinary Sachs-Wolfe effect, the velocity term is the Doppler CMB anisotropy at last scattering, and the integral is the Integrated Sachs-Wolfe effect produced by evolving potentials.

The line-of-sight solution for free-streaming photons follows the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-\hat{\mathbf{p}}({\eta }_f-\eta )$. If $\Phi =\Psi$ is spatially homogeneous, its gradient vanishes and $d\log ϵ/d\eta ={\Phi }^{\prime }$. Integration along the ray givesThe needed initial condition is the temperature pattern on the initial hypersurface; for a superhorizon adiabatic mode during matter domination one may use ${\Theta }_0({\eta }_i)=-2\Psi ({\eta }_i)/3$.

This calculation contains free streaming of the initial pattern and the gravitational temperature shift caused by the evolving potential, the homogeneous limit of the Integrated Sachs-Wolfe effect. A perfectly homogeneous potential changes only the unobservable sky monopole and therefore contributes no CMB angular power spectrum for $\ell \ge 1$. Direction-dependent Sachs-Wolfe effect, Doppler, acoustic, polarization, lensing, and rescattering contributions require spatially varying perturbations or collision terms and are outside this special calculation.