The Bose-Einstein distribution for photons is given by $$f(\textbf{x},p,\hat{p},t) = \frac{1}{e^{\frac{p}{T(t)[1+\Theta(\textbf{x},\hat{p},t)]}}-1}$$

where $p$ is the magnitude of the momentum of the photon, $T$ is the temperature and $\Theta$ is the fractional temperature perturbation. The chemical potential is zero.

This has been expanded to first order in $\Theta$ as follows:

$$f(\textbf{x},\textbf{p},t) = \frac{1}{e^{\frac{p}{T(t)}}-1} + \frac{\partial}{\partial T}[\frac{1}{e^{\frac{p}{T(t)}}-1}]T(t)\Theta(\textbf{x},\hat{p},t)$$

This doesn't look like a Taylor expansion. Could someone explain how this expansion has been made and also the general form of such an expansion.

  • $\begingroup$ Taylor it to the 1st order for $\Theta$. Once you have the result, you can compare the expressions and see that you'll get exactly what you provided in your question once you take the derivative wrt T. $\endgroup$
    – IcyOtter
    Jul 17 at 12:05

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