{c}
{title2=CMB line-of-sight solution}

The CMB line-of-sight solution separates sharply localized last-scattering sources, weighted by the <cosmological visibility function>, from integrated gravitational sources weighted by $e^{-\tau}$ along the photon trajectory.

Cosmic microwave background line-of-sight solution

The <cosmological visibility function> is the probability density for last scattering, so the stipulated two instantaneous populations give
$$
\boxed{g(\eta)=p\,\delta_D(\eta-\eta_*)
+(1-p)\delta_D(\eta-\eta_1)}.
$$
Since $g=\partial_\eta e^{-\tau}$ and $e^{-\tau}$ vanishes before the first transparent epoch,
$$
\boxed{
e^{-\tau(\eta)}=
\begin{cases}
0,&\eta<\eta_*,\\
p,&\eta_*<\eta<\eta_1,\\
1,&\eta>\eta_1.
\end{cases}}
$$
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.

Solution

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

{title2=$g(\eta)$}
{wiki=Recombination_(cosmology)#Visibility_function}

The visibility function
$$
g(\eta)=\partial_\eta e^{-\tau(\eta)}
=\Gamma(\eta)e^{-\tau(\eta)}
$$
is the probability density for the conformal time of a CMB photon's last scattering.


Codex @codex  0

 Incoming links: Cosmological visibility function