What Redshift Occurs When Photon and Baryon Densities Equalize at Recombination?

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SUMMARY

The discussion centers on calculating the redshift at recombination (z_{rec}) when the number density of photons (n_{\gamma}(z_{rec})) equals the number density of baryons (n_{b}(z_{rec})). Participants utilize the relationship \(\Omega_{\gamma}E_{bary} / hf_{mean}\Omega_{bary} = 1\) to derive this equality. However, there is confusion regarding the integration of blackbody radiation and the Cosmic Microwave Background (CMB) temperature into the calculations. The key challenge is determining how to equate \(\Omega_{\gamma}\) with the CMB temperature.

PREREQUISITES
  • Understanding of cosmological parameters such as baryon density (\(\Omega_{bary}\)) and photon density (\(\Omega_{\gamma}\))
  • Knowledge of blackbody radiation principles and the Planck distribution
  • Familiarity with the Cosmic Microwave Background (CMB) and its temperature
  • Basic concepts of ionization energy, particularly for hydrogen
NEXT STEPS
  • Study the Planck distribution for blackbody radiation and its implications for photon density
  • Learn about the relationship between CMB temperature and photon density in cosmology
  • Investigate the calculation of redshift in cosmological models
  • Explore the concept of ionization energy and its role in photon interactions with hydrogen
USEFUL FOR

Astronomy students, cosmologists, and physicists interested in the early universe, particularly those studying recombination and the evolution of cosmic structures.

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Homework Statement



We are to assume the recombination happens at redshift z_{rec} when the number density of photons n_{\gamma}(z_{rec}) capable of ionizing hydrogen is exactly equal to the number density of baryons n_{b}(z_{rec}). Use the measured number density of baryons, the temperature of the CMB and the blackbody radiation to find out at what redshift z_{rec} we have n_{\gamma}(z_{rec}) = n_{b}(z_{rec}).


2. The attempt at a solution
Sorry I don't have time to write in detail what I have so far. But basically I come down to \Omega_{\gamma}E_{bary} \over hf_{mean}\Omega_{bary} = 1

However this doesn't make use of blackbody radation, or the temperature of the CMB, if I'm supposed to equate \Omega_{\gamma} to Temperature of the CMB, then I'm not sure how to do that step.

Thanks
 
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Forget cosmology for a while.

If E_I is the ionization energy of hydrogen, any photon with energy greater than E_I can ionize hydrogen. Now consider a blackbody at temperature T. What is the number density for photons with energies greater than E_I?
 
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