The Cosmic Background Radiation — Question 3

Question 3

Radiation energy density carries four powers of temperature

Use $u=a_RT^4$ with $a_R=7.566\times10^{-16}$ J m$^{-3}$ K$^{-4}$ and T=2.725 K. Let c=$3.00\times10^8$ m/s.

1. Find energy density and equivalent mass density u/c squared.

2. At z=9 find the energy-density ratio and explain its four expansion factors.

Question 3 - Solution

Radiation energy density carries four powers of temperature

(a) Method and calculation

$$\boxed{u=7.566\times10^{-16}(2.725)^4\simeq4.17\times10^{-14}\ {\mathrm{J/m^3}}}.$$$$\boxed{\rho_\gamma=u/c^2\simeq4.64\times10^{-31}\ {\mathrm{kg/m^3}}}.$$

(b) Interpretation and further calculation

$$\boxed{u(z=9)/u_0=(1+9)^4=10^4}.$$$$u=n_\gamma\langle E\rangle,\quad n_\gamma\propto a^{-3},\quad\langle E\rangle\propto a^{-1}.$$

Physical check

Radiation loses an additional energy factor compared with nonrelativistic rest-mass density.