Photon energy density (${\rho }_{\gamma }=\int d^{3}pEf$)

The temporal-temporal component of the photon stress-energy tensor in a local orthonormal tetrad. For an isotropic one-particle distribution function expressed using comoving energy $ϵ=aE$, it is ${\bar{\rho }}_{\gamma }=4\pi a^{-4}\int dϵ{ϵ}^3\bar{f}(ϵ)$. A thermal photon temperature perturbation gives a first-order directional energy perturbation four times its fractional temperature change. Boundary conditions on the distribution justify this factor through integration by parts.