= Cosmological spectral flux-density relation
{title2=$F_{\nu_0}=(1+z)L_{(1+z)\nu_0}/(4\pi d_L^2)$}
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 $L_{\nu_e}/[4\pi a_0^2r^2(1+z)]$, with $\nu_e=(1+z)\nu_0$. Substituting <luminosity distance> $d_L=(1+z)a_0r$ 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>.
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