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Use one Fourier mode, with $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{n}}$ and unperturbed ray $\mathbf{x}(\tau )=\hat{\mathbf{n}}({\tau }_0-\tau )$. Put $E(\tau )=e^{ik\mu ({\tau }_0-\tau )}$, and let ${\tau }_{\ast }$ denote last scattering. In the supplied branch $h^{\prime }=h_s^{\prime }$, the scalar decomposition givesSince $E^{\prime \prime }=-k^2{\mu }^{2}E$, two integrations by parts give the exact endpoint decomposition of the metric contribution:For the matter-era growing mode, $h_s$ is quadratic in $\tau$, so the last integral vanishes. With $E^{\prime }=-ik\mu E$, the emission endpoint isand the observer endpoint is $-h_s^{\prime \prime }({\tau }_0)/(2k^2)-i\mu h_s^{\prime }({\tau }_0)/(2k)$. The observer terms are only a angular monopole and angular dipole; they are removed when studying the anisotropy multipoles $\ell \ge 2$.

For an adiabatic growing mode outside the horizon at last scattering, ${\delta }_{\gamma }=4{\delta }_m/3$, ${\delta }_m=-h/2$ after fixing the constant trace convention, and $h\propto {\tau }^2$. Hence the intrinsic emission term is smaller than $h_s^{\prime \prime }/(2k^2)$ by order $(k{\tau }_{\ast }{)}^2$. The derivative emission endpoint is smaller by order $k{\tau }_{\ast }$. Stress-energy conservation also makes the peculiar emission velocity subleading; its leading synchronous contribution cancels in ${\delta }_{\gamma }^{\prime }+2h^{\prime }/3$, with gradients supplying the remaining small term. Keeping the Fourier phase, which need not be small over the observer's distance, givesHere $\Theta =\delta T/T$, and the sum over modes is understood. The derivative is the second derivative of $h_s$, as printed in the original PDF. This approximation assumes matter-era growing adiabatic modes and neglects late evolving potentials; large wavelength alone does not remove the last integral in a general cosmology. The equality $h^{\prime }=h_s^{\prime }$ is used as the supplied synchronous condition, rather than as a general identity for every fluid solely from vanishing anisotropic stress.