- #1
Orion1
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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"
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"
http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation"
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