# CMB Temperature - Will hydrogen be ionized?

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1. May 25, 2015

### unscientific

1. The problem statement, all variables and given/known data
Energy density of radiation $\rho_r = 8 \times 10^{-14} J m^{-3}$ and energy density of matter $\rho_m = 2.63 \times 10^{-10} J m^{-3}$. Baryon density parameter is $\Omega_b = 0.04$. Temperature of CMB today is $2.73 K$. Ionization energy of Hydrogen is $13.6 eV$.

(a)Find the scale factor and temperature of radiation when the two energy densities are balanced.
(a)Find photons to baryon ratio.
(b)Would hydrogen have been ionized?

2. Relevant equations

3. The attempt at a solution

Part(a)

For matter and radiation energy density to balance, we need $\rho_r a^{-4} = \rho_m a^{-3}$, so we have
$$a = \frac{\rho_r}{\rho_m} = 3 \times 10^{-4}$$
Temperature varies inversely with scale factor:
$$T = \frac{T_0}{a} = 9100 K$$

Part(b)

Photon energy density is given by $\rho = \frac{\pi^2}{15} (k_B T) \left( \frac{k_B T}{\hbar c} \right)^3$. Energy per photon is given by $k_B T$. Thus number density of photon is $n_\gamma = \frac{\rho}{k_B T} = 3.7 \times 10^8$. Given $\Omega_b = 0.04$, energy density of baryon is $3.41 \times 10^{-11}$. Energy of baryon is typically mass of neutron or proton $938 MeV$. Thus number density of baryon is $n_b = 0.23$. Ratio is
$$\frac{n_\gamma}{n_b} = 10^9$$

Part (c)
To ionize hydrogen, we require a temperature of $T = \frac{E}{k_B} = 158 000 K$. But that doesn't seem right..

I feel like im a factor of 20 off..

2. May 27, 2015

### Khashishi

Try using the Saha ionization equation for part c. There should be a decrease in the temperature requirement since ionization causes an increase in entropy.

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