Photon angular temperature moments (${\delta }_{\gamma }=4⟨\Theta ⟩,v_{\gamma }^i=3⟨\Theta e^i⟩$)

Let $⟨F⟩=(4\pi {)}^{-1}\int Fd\Omega$ and $f=\bar{f}(ϵ)-ϵ{\bar{f}}^{\prime }(ϵ)\Theta (\mathbf{e})$ with comoving energy $ϵ=aE$. If ${ϵ}^4\bar{f}$ vanishes at both integration endpoints, integration by parts gives $-\int {ϵ}^4{\bar{f}}^{\prime }dϵ=4\int {ϵ}^3\bar{f}dϵ$. The kinetic stress-energy tensor then gives ${\delta }_{\gamma }=4⟨\Theta ⟩$, $v_{\gamma }^i=3⟨\Theta e^i⟩$, $\delta P=\bar{\rho }{\delta }_{\gamma }/3$, and ${\pi }^{ij}=4\bar{\rho }⟨\Theta (e^i{e}^j-{\delta }^{ij}/3)⟩$. If $\Pi =-\pi$ is used, the last expression changes sign. Frequency independence of $\Theta$ is essential to this common temperature description.

A direction-dependent fractional photon temperature change. For a thermal spectrum, $f(ϵ)=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ is its linear expansion at fixed comoving energy. For a nonthermal spectrum the same ansatz is a brightness dilation, rather than necessarily a thermodynamic temperature. In unweighted Legendre polynomial multipoles, $\Theta ={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }$, the photon density, velocity and anisotropic stress are ${\delta }_{\gamma }=4{\Theta }_0$, $v_{\gamma }=-{\Theta }_1$ and ${\Pi }_{\gamma }=-3{\Theta }_2/5$.