Fitting the Cosmic Microwave Background

Question

The NASA Cosmic Background Explorer (COBE) satellite carried an instrument, FIRAS (Far-Infrared Absolute Spectrophotometer) to measure the cosmic microwave background (CMB) radiation, which was confirmed to be distributed according to a black-body curve in accordance with the big bang theory:

\begin{align*} I(\tilde{\nu}, T) = \frac{2h\tilde{\nu}^3c^2}{\exp\left(\frac{hc\tilde{\nu}}{k_\mathrm{B}T}\right) - 1} \end{align*} where the radiation frequency is expressed in wavenumbers, $\mathrm{cm^{-1}}$, and the speed of light, $c$, is taken to be in $\mathrm{cm\,s^{-1}}$.

The data file cmb-data.txt contains measurements of $I(\tilde{\nu})$ based on the FIRAS observations. Note that the units of $I$ in this file are $\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}\,cm}$ and that $1\;\mathrm{J}\equiv 10^7\;\mathrm{erg}$. Use scipy.optimize.curve_fit to determine the temperature of the CMB and take the estimated $1\sigma$ error in the measurement to be $2 \times 10^{-6}\;\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}\,cm}$.


Solution

To access solutions, please obtain an access code from Cambridge University Press at the Lecturer Resources page for my book (registration required) and then sign up to scipython.com providing this code.