Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 4

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

The cosmological visibility function is the probability density for last scattering, so the stipulated two instantaneous populations giveSince $g={\partial }_{\eta }{e}^{-\tau }$ and $e^{-\tau }$ vanishes before the first transparent epoch,Its plot is a nondecreasing step function: it jumps from zero to $p$ at recombination and from $p$ to one at ${\eta }_1$. The cosmological optical depth interpretation is that $e^{-\tau (\eta )}$ is the probability that a photon present at time $\eta$ reaches the observer without any later scattering.

Write ${\mathbf{x}}_j={\mathbf{x}}_0+({\eta }_0-{\eta }_j)\hat{\mathbf{n}}$. The delta functions in $g$ evaluate the local last-scattering sources, while the piecewise $e^{-\tau }$ weights the integrated gravitational source. ThusThe first two lines are weighted Sachs-Wolfe combination and Doppler sources at the two last-scattering surfaces; the final lines are the corresponding Integrated Sachs-Wolfe effect.

To predict these quantities from primordial perturbations, one solves the coupled linear Einstein field equations and Boltzmann equation hierarchy for photons, baryons, cold dark matter, and neutrinos, together with the ionization history that determines $\Gamma$ and hence $g$. The solutions provide ${\Theta }_0$, ${\mathbf{v}}_e$, $\Phi$, and $\Psi$ at both surfaces and along the line of sight.