 Solution

To first order, propagate along the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e}$ and keep its orthonormal tetrad direction fixed. Corrections to direction or trajectory multiply an already first-order temperature source and are second order. Define the angular scattering sourceUsing the photon energy redshift from a tensor metric perturbation, the Photon Boltzmann equation with Thomson scattering along this ray becomesMultiply by the survival probability $e^{-\tau }$. The cosmological visibility function is $g(\eta )=-\dot{\tau }{e}^{-\tau }$, so integration givesEvery source here is evaluated on the same ray, and $\tau ({\eta }_0)=0$.

The instantaneous-visibility approximation makes $e^{-\tau }$ a step from zero to one at ${\eta }_{\ast }$. It suppresses the early boundary term and localizes the scattering source to $S_{\ast }$. To obtain only the requested tensor CMB line-of-sight source, additionally neglect this last-scattering angular source: in leading tight coupling, the tensor-induced incident quadrupole is small, and a tensor mode has no scalar monopole or dipole. Ignore later reionization and its scattering as well. ThenInstantaneous visibility alone would leave the term $S_{\ast }$; neglect of that term is an additional approximation. The time derivative on $h$ is essential and is present in the PDF, although it was lost in the extracted TeX.