The Cosmic Background Radiation — Question 2

Question 2

Frequency and wavelength spectra peak at different transformed locations

For T=2.725 K use $\lambda_{\max}T=2.898\times10^{-3}$ m K for $B_\lambda$, and $\nu_{\max}/T=5.879\times10^{10}$ Hz/K for $B_\nu$. Take c=$3.00\times10^8$ m/s.

1. Calculate both peak locations.

2. Calculate $c/\lambda_{\max}$ and compare with $\nu_{\max}$; explain using the spectral-density Jacobian.

Question 2 - Solution

Frequency and wavelength spectra peak at different transformed locations

(a) Method and calculation

$$\boxed{\lambda_{\max}=2.898\times10^{-3}/2.725=1.06349\ {\mathrm{mm}}}.$$$$\boxed{\nu_{\max}=5.879\times10^{10}(2.725)=160.203\ {\mathrm{GHz}}}.$$

(b) Interpretation and further calculation

$$\boxed{c/\lambda_{\max}=282.091\ {\mathrm{GHz}}\ne\nu_{\max}}.$$$$B_\nu\,|d\nu|=B_\lambda\,|d\lambda|,\quad\boxed{B_\lambda=B_\nu\,c/\lambda^2}.$$

Physical check

The same spectrum has different maxima when expressed per unit frequency and per unit wavelength; directly converting one peak does not locate the other.