Wednesday, November 20, 2013

radiation density

can you express the density parameter $\Omega_\gamma$ of a Planck-distributed relativistic species as a function of temperature only?

1 comment:

  1. in fact, the energy density $\rho$ of a Planck-distributed quantum gas is just $\sigma T^4$ where $\sigma=\pi k_B^4/(15(\hbar c)^3)$ is the Stefan-Boltzmann constant and $T$ is the temperature. this can be expressed in units of the critical density $\rho_\mathrm{crit}=3H_0^2/(8\pi G)$:
    \begin{equation}
    \Omega_\gamma = \frac{\rho}{\rho_\mathrm{crit}} = \frac{8\pi G \sigma}{3H_0^2}T^4 = \frac{8\pi^2}{45}\frac{Gk_B^4}{H_0^2(\hbar c)^3}T^4
    \end{equation}
    with an awesome collection of natural constants.

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