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unscientific

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## 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

[tex]a = \frac{\rho_r}{\rho_m} = 3 \times 10^{-4}[/tex]

Temperature varies inversely with scale factor:

[tex]T = \frac{T_0}{a} = 9100 K[/tex]

__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

[tex]\frac{n_\gamma}{n_b} = 10^9[/tex]

__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..