• Introduction
• Introduction
• Q1) Recombination
  • Short Answer
  • Notes from Barth's Cosmology (2019)
• Q2) Flat Universe Properties

What is recombination? At what temperature did it occur? Explain why this does not match the ionization potential of hydrogen.

Recombination refers to the time at which the temperature of the early Universe became cool enough such that it was thermodynamically favorable for the ionized plasma of free electrons and ions to couple and form neutral atoms. Numerically, this might be defined as the moment when the number density of ions is equal to the number density of neutral atoms.

Temperature: ${\mathbf T \sim 1,000,{\rm K}}$ (corresponding to an energy of $\sim 0.1,{\rm eV}$) at a redshift of $z \sim 1,100$.

This does not match the ionization potential of hydrogen because the early Universe (as it was hot and dense) can be described by a blackbody with a characteristic distribution of photon energies including an exponential tail of high energy photons (Wein's tail). While the peak of the blackbody spectrum describing the temperature of the early Universe is below the ionization energy of hydrogen, the photons in the high-energy exponential tail of the blackbody spectrum have sufficient energies for photoionization.

Steps: Integrate blackbody to find what percentage of photons have energies higher than 13.6 eV. Extrapolate measured $\Omega_b$ back in time to get the photon to baryon ratio. The number of photons above 13.6 eV equals number of baryons at 5600 K.

$$ X_p = \frac{N_p}{N_p + N_H} \approx 0.1 $$ at $T_{rec}$.

Saha equation:

$$\frac{X^2}{1-X} = \frac{(2\pi m_e kT)^{3/2}}{(n_e + n_H)(2\pi h)^3} e^{-13.6eV/kT}$$

At what temperature is $X$ 10%? Depends on $\Omega_b h^2$ because of the $(n_e + n_H)$ in denominator. Higher density of baryons makes recombination happen earlier. Solving for when $X_p$ is 0.1 gives $T\sim 3600K$ or 0.3 eV. This is recombination.

Then ask, when will the universe be 'transparent'? Can define that by when optical depth $\tau = 1$. This comes out to be $T\sim3200$K, $z\sim1100$. This is decoupling.

Links: - [Jessica Campbell Qual Notes](./cosmo_extragal/Campbell/cosmology.pdf)