Photon brightness equation for a negative spatial perturbation (${\Delta }^{\prime }+ik\mu \Delta =2h_{ij}^{\prime }{n}^i{n}^j$)

For synchronous gauge metric $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}-h_{ij})d{x}^{i}d{x}^j]$, conserved-background comoving momentum obeys $q^{\prime }=q{h}_{ij}^{\prime }{n}^i{n}^j/2$. In the linear Collisionless Boltzmann equation, insert $\Delta =-4f_1/(q{f}_0^{\prime })$ to obtain the displayed source. Using ${\delta }_{ij}+h_{ij}$ instead reverses both signs; it does not change the physics. The scalar contraction is $h_{ij}^{\prime }{n}^i{n}^j=(h^{\prime }-h_s^{\prime })/3+{\mu }^2{h}_s^{\prime }$.