= Angular average
{title2=$\langle F\rangle_\Omega=\frac1{4\pi}\int_{S^2}F\,d\Omega$}
The normalized <angular average> is over the unit <sphere>, with <solid angle> measure $d\Omega$. Rotational symmetry gives $\langle e^i\rangle_\Omega=0$, $\langle e^ie^j\rangle_\Omega=\delta^{ij}/3$, and $\langle e^ie^je^ke^\ell\rangle_\Omega=(\delta^{ij}\delta^{k\ell}+\delta^{ik}\delta^{j\ell}+\delta^{i\ell}\delta^{jk})/15$. The constants follow by contracting indices and using $e^ie_i=1$. These identities extract <photon angular temperature moments>. For normalized <spherical harmonics>, $\int Y_{\ell m}^*Y_{\ell' m'}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}$; their <angular average> instead has an extra factor $1/(4\pi)$.
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