= 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 source
$$
S(\eta,\mathbf x,\mathbf e)=\frac{3}{16\pi}\int d\Omega_{\mathbf m}\,\Theta(\eta,\mathbf x,\mathbf m)[1+(\mathbf e\cdot\mathbf m)^2].
$$
Using the <photon energy redshift from a tensor metric perturbation>, the <Photon Boltzmann equation with Thomson scattering> along this ray becomes
$$
\frac{d\Theta}{d\eta}-\dot\tau\Theta=-\frac12\dot h_{ij}e^ie^j-\dot\tau S.
$$
Multiply by the survival probability $e^{-\tau}$. The <cosmological visibility function> is $g(\eta)=-\dot\tau e^{-\tau}$, so integration gives
$$
\Theta_0=e^{-\tau_i}\Theta_i+\int_{\eta_i}^{\eta_0}gS\,d\eta-\frac12\int_{\eta_i}^{\eta_0}e^{-\tau}\dot h_{ij}e^ie^j\,d\eta.
$$
Every 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 $\eta_*$. It suppresses the early boundary term and localizes the scattering source to $S_*$. 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. Then
$$
\boxed{\Theta(\eta_0,\mathbf x_0,\mathbf e)\simeq-\frac12\int_{\eta_*}^{\eta_0}\dot h_{ij}(\eta,\mathbf x_0-(\eta_0-\eta)\mathbf e)e^ie^j\,d\eta.}
$$
\b[Instantaneous visibility alone would leave the term $S_*$; 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.
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