Cosmological background intensity from comoving emissivity ($I_{{\nu }_0}=\frac{c}{4\pi }\int \frac{{ϵ}_{(1+z){\nu }_0}(z)}{(1+z)H(z)}dz$)

Let ${ϵ}_{{\nu }_e}(z)$ 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: $d{V}_c=r^2(c/H)dzd\Omega$, whereas $F_{{\nu }_0}=(1+z)L_{{\nu }_e}/[4\pi (1+z{)}^2{r}^2]$. Absorption inserts a transmission factor, and evolving sources are represented by the redshift dependence of $ϵ$.

For nonevolving comoving number density $n_0$, identical spectra $L_{\nu }\propto {\nu }^{\alpha }$, and no attenuation, the cosmological background intensity from comoving emissivity in an Einstein-de Sitter universe is proportional to ${\int }_0^{\infty }(1+z{)}^{\alpha -5/2}dz$. The improper integral is finite exactly for $\alpha <3/2$, with value $1/(3/2-\alpha )$. Equality gives logarithmic divergence; larger $\alpha$ gives a power-law divergence. A formation epoch or a spectral break can regularize this idealized model.