Redshift at recombination

[utopiaNow]

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

 

[George Jones]

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 $E_I$?