Cosmological bolometric background from conserved sources (${\mathcal{F}}_0=cL{n}_0\int a(t)dt$)

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 $3cL{n}_0{t}_0/5$.