Optical Depth

In this post, I calculate the Thompson scattering optical depth for the DREaM Catalog

Calculation

Thompson Optical Depth (from eq6 of Naidu et al.):

\[\tau = c \bar{n_H} \sigma_T \int_0^z f_e Q_{H_{II}} \dfrac{(1+z')^2}{H(z)} dz'\]

where

\(c\): speed of light
\(\bar{n}_H\): comoving hydrogen density (\(1.9 \times 10^{-7} \rm{cm}^3\))
\(\sigma_T\): electron Thomson scattering cross-section (\(6.6524587158 \times 10^{-29 \rm{m}^2}\))
\(f_e\) = number of free electrons for every hydrogen nucleus in the ionized IGM, \(f_e = (1+(1−X)/4X)\)
\(Q_{H_{II}}\): Ionized fraction (calculated as here)
\(H(z)\): Hubble parameter

Here is the plot, compared to the Planck 2019 data:

Looks good!