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CBR photon density at equilibrium temperature

  1. Aug 30, 2010 #1

    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]
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 12, 2010 #2
    Last edited by a moderator: Apr 25, 2017
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