# CBR photon density at equilibrium temperature

1. Aug 30, 2010

### Orion1

CBR photon equilibrium temperature:
$$T_{\gamma} = 2.725 \; \text{K}$$

CBR photon density at equilibrium temperature $$T_{\gamma}$$:
$$n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{e^x - 1} dx$$

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

Integration:
$$\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \sum_{n=1}^\infty n^{-3} = 2 \zeta(3)$$

Complex variable:
$$\boxed{s = 3}$$

$$\boxed{\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \zeta(3)}$$

Where $$\zeta(s)$$ is the Riemann zeta function.

Riemann zeta function:
$$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$

The CBR photon density at equilibrium temperature $$T_{\gamma}$$:
$$\boxed{n_{\gamma} = \frac{2 \zeta(3)}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3}$$

$$\boxed{n_{\gamma} = 4.105 \cdot 10^{8} \; \frac{\text{photons}}{\text{m}^3}}$$

Are these equations correct?

Reference:
http://en.wikipedia.org/wiki/Riemann_zeta_function" [Broken]
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"