Scalar photon energy redshift
= Scalar photon energy redshift
{title2=$d\ln\epsilon/d\eta=\phi^{\prime}-\mathbf e\cdot\nabla\psi$}
For the <Newtonian gauge in cosmology> with potentials $\psi,\phi$, photon comoving momentum $\epsilon$ obeys $d\ln\epsilon/d\eta=\phi'-\mathbf e\cdot\nabla\psi=-d\psi/d\eta+\phi'+\psi'$. The homogeneous physical-energy redshift is separated by $E=\epsilon/a$. The total derivative follows the unperturbed photon path at linear order.