Synchronous photon brightness equation
= Synchronous photon brightness equation
{title2=$\Delta'+ik\mu\Delta=-2h'_{ij}n^in^j$}
For collisionless photons in <synchronous gauge>, their perturbed momentum obeys $q'=-qh'_{ij}n^in^j/2$. Terms in $q'\partial_qf_1$ and direction deflection times angular derivatives of $f_1$ are second order. Integrating the remaining linear <Collisionless Boltzmann equation> over $q^3dq$ gives the displayed equation, because $\int q^4f_0' dq=-4\int q^3f_0dq$.
// Target: cosmology.bigb