= Cosmological background intensity from comoving emissivity
{title2=$I_{\nu_0}=\dfrac{c}{4\pi}\int\dfrac{\epsilon_{(1+z)\nu_0}(z)}{(1+z)H(z)}\,dz$}
Let $\epsilon_{\nu_e}(z)$ be the <comoving spectral emissivity>, namely total emitted <spectral luminosity> per <comoving volume>. In a homogeneous transparent <FRW metric>, a radial shell contributes its <comoving volume> times the <cosmological spectral flux-density relation>. Its transverse-area factors cancel, leaving the displayed integral. In a flat model this cancellation is explicit: $dV_c=r^2(c/H)dz\,d\Omega$, whereas $F_{\nu_0}=(1+z)L_{\nu_e}/[4\pi(1+z)^2r^2]$. Absorption inserts a transmission factor, and evolving sources are represented by the redshift dependence of $\epsilon$.
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