 Solution

Let $\hat{\mathbf{n}}$ point from the observer toward the emission point, so the photon propagation direction is $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ andAlong this path $D=d/d\eta ={\partial }_{\eta }-\hat{\mathbf{n}}\cdot \nabla$. Substituting $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ into the Photon Boltzmann equation with Thomson scattering givesBecause ${\partial }_{\eta }{e}^{-\tau }=\Gamma e^{-\tau }$, multiplication by the integrating factor giveswhereIntegrating along the ray and using $e^{-\tau }\to 0$ at sufficiently early time yields the Cosmic microwave background line-of-sight solution