Cosmological spectral flux-density relation ($F_{{\nu }_0}=(1+z)L_{(1+z){\nu }_0}/(4\pi d_L^2)$)

A comoving source's spectral luminosity $L_{{\nu }_e}$ specifies emitted power per unit emitted frequency. The cosmological redshift reduces photon energy and arrival rate by two factors of $1+z$, but also narrows the received frequency interval by one factor. Consequently its observed spectral flux density is $L_{{\nu }_e}/[4\pi a_0^2{r}^2(1+z)]$, with ${\nu }_e=(1+z){\nu }_0$. Substituting luminosity distance $d_L=(1+z)a_{0}r$ gives the displayed relation. Integrating over received frequency recovers the bolometric radiative flux $L/(4\pi d_L^2)$. A fixed-frequency comparison must evaluate the source spectrum at the shifted emitted frequency.