# 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

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