CMB Temperature - Will hydrogen be ionized?

[unscientific]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

Part(a)[/B]
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 I am a factor of 20 off..

 

[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.