# CMB Temperature - Will hydrogen be ionized?

• unscientific
In summary, the energy density of radiation and matter are given by ##\rho_r = 8 \times 10^{-14} J m^{-3}## and ##\rho_m = 2.63 \times 10^{-10} J m^{-3}## respectively. The baryon density parameter is ##\Omega_b = 0.04## and the temperature of CMB today is ##2.73 K##. The ionization energy of Hydrogen is ##13.6 eV##. To balance the two energy densities, the scale factor is ##a = 3 \times 10^{-4}## and the temperature of radiation is ##T = 9100 K##. The photons to b
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?

## 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..

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.

## 1. What is CMB temperature?

The CMB (cosmic microwave background) temperature is the temperature of the radiation left over from the Big Bang, which permeates the entire universe.

## 2. How is CMB temperature measured?

CMB temperature is measured through observations of the cosmic microwave background radiation using specialized telescopes and detectors.

## 3. Why is the CMB temperature important?

The CMB temperature provides crucial evidence for the Big Bang theory and helps us understand the early stages of the universe's evolution. It also gives us insight into the composition and properties of the universe.

## 4. What does the CMB temperature have to do with hydrogen being ionized?

As the universe expanded and cooled after the Big Bang, the CMB temperature also decreased. When the CMB temperature reached around 3000K, it allowed for the formation of neutral hydrogen atoms. If the CMB temperature were to increase, it could potentially cause the reionization of hydrogen atoms, turning them into ions.

## 5. Will the CMB temperature ever reach a point where hydrogen is completely ionized?

It is currently believed that the CMB temperature will continue to decrease due to the expansion of the universe, so it is unlikely that hydrogen will ever be completely ionized. However, there are still ongoing studies and research to better understand the evolution of the CMB temperature and its effects on the ionization of hydrogen.

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