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CBR photon equilibrium temperature:

[tex]T_{\gamma} = 2.725 \; \text{K}[/tex]

CBR photon density at equilibrium temperature [tex]T_{\gamma}[/tex]:

[tex]n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{e^x - 1} dx[/tex]

This is my solution for the integration equation on reference 2.

Integration:

[tex]\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \sum_{n=1}^\infty n^{-3} = 2 \zeta(3)[/tex]

Complex variable:

[tex] \boxed{s = 3}[/tex]

[tex]\boxed{\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \zeta(3)}[/tex]

Where [tex]\zeta(s)[/tex] is the Riemann zeta function.

Riemann zeta function:

[tex]\zeta(s) = \sum_{n=1}^\infty n^{-s}[/tex]

The CBR photon density at equilibrium temperature [tex]T_{\gamma}[/tex]:

[tex]\boxed{n_{\gamma} = \frac{2 \zeta(3)}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3}[/tex]

[tex]\boxed{n_{\gamma} = 4.105 \cdot 10^{8} \; \frac{\text{photons}}{\text{m}^3}}[/tex]

Are these equations correct?

Reference:

http://en.wikipedia.org/wiki/Riemann_zeta_function" [Broken]

http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"

http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation" [Broken]

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# CBR photon density at equilibrium temperature

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