= Solution
The <Collisionless Boltzmann equation> is $df/d\eta=0$. At first order, directional deflections act only on the isotropic background and hence give no angular contribution. Differentiating $f=\bar f-\Theta\,d\bar f/d\ln\epsilon$ along the unperturbed trajectory gives
$$
0=\frac{d\bar f}{d\ln\epsilon}
\left[\frac{d\ln\epsilon}{d\eta}-
(\partial_\eta+\mathbf e\cdot\nabla)\Theta\right].
$$
Using the tensor redshift of the comoving energy,
$$
\boxed{\dot\Theta+\mathbf e\cdot\nabla\Theta
=-\frac12\dot h_{ij}e^ie^j}.
$$
The <method of characteristics> follows the backward ray $\mathbf x(\eta')=\mathbf x-(\eta-\eta')\mathbf e$. With zero initial perturbation, the line-of-sight solution is
$$
\boxed{\Theta(\eta,\mathbf x,\mathbf e)
=-\frac12\int_0^\eta d\eta'\,
\dot h_{ij}\bigl(\eta',\mathbf x-(\eta-\eta')\mathbf e\bigr)e^ie^j}.
$$
To this order the frame and coordinate direction components may be identified inside a quantity already proportional to $h$.
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