The Cosmic Background Radiation — Question 1

Question 1

The microwave background cools as its wavelengths stretch

Assume freely propagating blackbody radiation with $T(z)=T_0(1+z)$ and $T_0=2.725$ K. Consider an epoch z=999.

1. Find the temperature and scale factor at that epoch.

2. A photon observed today at wavelength 1.00 mm was emitted then. Find its earlier wavelength and the ratio of its earlier energy to its present energy.

Question 1 - Solution

The microwave background cools as its wavelengths stretch

(a) Method and calculation

$$\boxed{T=2.725(1000)=2725\ {\mathrm{K}}}.$$$$\boxed{a=1/(1+z)=.001}.$$

(b) Interpretation and further calculation

$$\boxed{\lambda_e=\lambda_0/1000=1.00\ \mu{\mathrm{m}}}.$$$$\boxed{E_e/E_0=\lambda_0/\lambda_e=1000}.$$

Physical check

An isotropically redshifted Planck distribution remains Planckian when temperature scales inversely with the scale factor.