Cosmological spectral background ($I_{{\nu }_0}$)

The cosmological spectral background is received radiant power per detector area, received frequency interval and solid angle, summed over a cosmological source population. Its intensity depends on emitted spectra, their cosmic evolution, the expansion history and absorption. A finite radial comoving distance alone does not guarantee a finite background when high-redshift sources are sampled at increasingly high emitted frequencies.

Normalize the scale factor to $a_0=1$. Identical isotropic sources of bolometric luminosity $L$ and conserved comoving number density $n_0=a^{3}n$ produce all-sky radiative flux ${\mathcal{F}}_0=cL{n}_0\int a(t)dt$ in a transparent FLRW metric. The source counts on an FLRW past light cone and cosmological flux dimming cancel the transverse shell-area factor. For arbitrary proper source density the expression is instead $cL\int n(t)a(t{)}^{4}dt$. The intensity per solid angle is ${\mathcal{F}}_0/(4\pi )$. A detector of collecting area $A$ receives power $A{\mathcal{F}}_0$ under an all-sky effective-area convention.

For an open matter-dominated Friedmann solution, put $u=H_0\sqrt{1-{\Omega }_0}\tau$, $a={\Omega }_0(\cosh u-1)/[2(1-{\Omega }_0)]$ and $\cosh u_0=2/{\Omega }_0-1$. The cosmological bolometric background from conserved sources has $J(u)=\sinh (2u)/4-2\sinh u+3u/2$, since $J^{\prime }=(\cosh u-1{)}^2$ and $dt=adu/[H_0\sqrt{1-{\Omega }_0}]$. At fixed $H_0$, the limit ${\Omega }_0\to 0$ is $cL{n}_0/(2H_0)$, whereas the Einstein-de Sitter universe gives $2cL{n}_0/(5H_0)$. Keeping the cosmic age fixed instead reverses this ordering: the two values are respectively $cL{n}_0{t}_0/2$ and $3cL{n}_0{t}_0/5$.

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.