= Solution
For two neighboring comoving points separated along the zeroth-order direction $\widehat{\mathbf n}$ by coordinate distance $\delta\chi$, the <synchronous gauge> spatial metric gives the <cosmological proper distance>
$$
\delta r=a(t)\delta\chi
\sqrt{(\delta_{ij}-h_{ij})\widehat n_i\widehat n_j}
=a(t)\delta\chi\left(1-\frac12h_{ij}\widehat n_i\widehat n_j\right)
+O(h^2).
$$
Differentiate with respect to <cosmic time> at fixed coordinate separation. To first order,
$$
\frac{d\,\delta r}{dt}
=\left(H-\frac12\dot h_{ij}\widehat n_i\widehat n_j\right)\delta r.
$$
A <photon> crosses this neighboring interval in $\delta t=\delta r$ when $c=1$, so its local relative recession velocity is
$$
\boxed{\Delta v\simeq
\left(\frac{\dot a}{a}-\frac12\dot h_{ij}\widehat n_i\widehat n_j\right)\delta t.}
$$
At each infinitesimal step, the local <Doppler effect> gives $d\log\nu=-\Delta v$. Therefore
$$
d\log\nu=-H\,dt+\frac12\dot h_{ij}\widehat n_i\widehat n_j\,dt.
$$
The first term integrates to the homogeneous <cosmological redshift>, $\bar\nu\propto a^{-1}$. Subtracting it isolates the metric contribution to the fractional frequency shift. A shifted <blackbody radiation> spectrum has its <temperature> shifted by the same fraction as its <photon> frequencies, giving
$$
\boxed{\left.\frac{\delta T}{T}\right|_{\rm metric}
=\frac{\delta\nu}{\bar\nu}
=\frac12\int_{t_{\rm dec}}^{t_0}
\dot h_{ij}\bigl(t,\mathbf x(t)\bigr)
\widehat n_i\widehat n_j\,dt.}
$$
The dot denotes a partial time derivative at fixed comoving position, evaluated along the zeroth-order <null geodesic>. It is not the total derivative of $h_{ij}$ along the ray, so the integral cannot in general be replaced by a difference of $h_{ij}$ at the endpoints. Deflection of the ray or its direction multiplies an already first-order metric perturbation and is second order here. This is the metric part of the <Sachs-Wolfe effect> in the negative-spatial-perturbation convention; an intrinsic emission <temperature> fluctuation or an emitter velocity would add separate boundary terms. The time derivative is present in the original PDF, although missing from the converted TeX's displayed integral.
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