= Photon angular temperature moments
{title2=$\delta_\gamma=4\langle\Theta\rangle,\quad v_\gamma^i=3\langle\Theta e^i\rangle$}
Let $\langle F\rangle=(4\pi)^{-1}\int F\,d\Omega$ and $f=\bar f(\epsilon)-\epsilon\bar f'(\epsilon)\Theta(\mathbf e)$ with comoving energy $\epsilon=aE$. If $\epsilon^4\bar f$ vanishes at both integration endpoints, <integration by parts> gives $-\int\epsilon^4\bar f'\,d\epsilon=4\int\epsilon^3\bar f\,d\epsilon$. The <kinetic stress-energy tensor> then gives $\delta_\gamma=4\langle\Theta\rangle$, $v_\gamma^i=3\langle\Theta e^i\rangle$, $\delta P=\bar\rho\delta_\gamma/3$, and $\pi^{ij}=4\bar\rho\langle\Theta(e^ie^j-\delta^{ij}/3)\rangle$. If $\Pi=-\pi$ is used, the last expression changes sign. Frequency independence of $\Theta$ is essential to this common temperature description.
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