# Expanding the Bose-Einstein distribution with fractional temperature perturbation

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.

• 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. Jul 17 at 12:05