= Photon brightness equation for a negative spatial perturbation
{title2=$\Delta'+ik\mu\Delta=2h'_{ij}n^in^j$}
For <synchronous gauge> metric $ds^2=a^2[-d\tau^2+(\delta_{ij}-h_{ij})dx^idx^j]$, conserved-background <comoving momentum> obeys $q'=qh'_{ij}n^in^j/2$. In the linear <Collisionless Boltzmann equation>, insert $\Delta=-4f_1/(qf_0')$ to obtain the displayed source. Using $\delta_{ij}+h_{ij}$ instead reverses both signs; it does not change the physics. The scalar contraction is $h'_{ij}n^in^j=(h'-h_s')/3+\mu^2h_s'$.
Back to article page
