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To match the positive source printed in the question, choose synchronous gauge metricThe commonly used convention with ${\delta }_{ij}+h_{ij}$ gives the opposite source sign; switching conventions means replacing $h$ and $h_S$ by their negatives together. This distinction is necessary when comparing with the usual synchronous photon brightness equation.

Let $p^{\alpha }=d{x}^{\alpha }/d\lambda$ be the photon tangent for an affine parameter. The covariant geodesic equation isFor $\nu =0$, the homogeneous scale-factor contributions cancel by the null geodesic condition, leaving $d{\tilde{p}}_0/d\lambda =-a^2{h}_{ij}^{\prime }{p}^i{p}^j/2$. Since ${\tilde{p}}_0=-a^2{p}^0=-q$ and $d\tau /d\lambda =p^0$, use the background $p^i=p^0{n}^i$ in this first-order term to obtainUsing the affine form also avoids confusing a covariant $p_0$ with the contravariant factor $p^0$ required when changing the geodesic parameter to coordinate time.

In conformal time, the equation from part(a) becomes $f_1^{\prime }+n^i{\partial }_i{f}_1+q^{\prime }{f}_0^{\prime }=0$. A Fourier mode proportional to $e^{i\mathbf{k}\cdot \mathbf{x}}$ has streaming term $ik\mu f_1$. Multiply by $-4/(q{f}_0^{\prime })$; its prefactor is background-time independent. With $\Delta =-4f_1/(q{f}_0^{\prime })$,Contracting the scalar decomposition explicitly givessoThis is the photon brightness equation for a negative spatial perturbation. For a perturbation of blackbody-temperature form, $f_1=-q{f}_0^{\prime }\Theta$, so $\Delta =4\Theta$ is the photon brightness perturbation. A more general initial spectral distortion can retain $q$ dependence; the gravitational forcing is still independent of $q$ in this linear massless equation.